Struct Multidisc Optim 23, 214–221 Springer-Verlag 2002 Digital Object Identifier (DOI) 10.1007/s00158-002-0179-1
Optimization of nonlinear trusses using a displacement-based approach S. Missoum, Z. G¨ urdal and W. Gu
Abstract A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust.
Key words geometric and material nonlinearities, twolevel optimization, linear programming, dual variables, truss design
Received December 4, 2000 Communicated by J. Sobieski S. Missoum, Z. G¨ urdal and W. Gu Aerospace and Ocean Engineering Department, Virginia Polytechnic Institute, Blacksburg, Virginia 24061, USA e-mail:
[email protected],
[email protected],
[email protected]
Parts of this research have been presented as paper 20001392 at the 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Atlanta, GA, April 3–6, 2000
1 Introduction Structural optimization has been a rapidly developing research field for the last three decades. Most of the research performed so far is based on the evaluation of a structural response and its sensitivities using the finite element method (FEM). In such an approach, crosssectional areas of rods, thicknesses of plate and shell elements are commonly used as the optimization design variables. A typical strategy for the solution of an optimization problem is, therefore, an iterative process in which the structural analysis equations are solved repeatedly until no progress can be made. Depending on the number of variables, such a technique may require a large number of analyses that are costly and time consuming, especially if the structural analysis is performed using the FEM. The cost of each analysis also depends on the type of analysis required to compute the desired responses. Typically, nonlinear analyses are the most demanding type of analysis and their use in structural optimization can be nearly impossible for large structures. In addition to their high computational cost, optimization of structures with nonlinear response has not received much attention because of the common assumption that a linear analysis based design is a conservative one. An interesting counter example has been found in the paper by Ringertz (1988), where a truss structure subjected to stress constraints with a linear design would fail if a more realistic nonlinear analysis were performed. Such examples underline the need for solution strategies for structural optimization problems with nonlinear response characteristics. One of the efficient approaches to solving optimization problems for nonlinear structures is to use an approach commonly referred to as the Simultaneous Analysis and Design (SAND) (Haftka and Kamat 1985). The approach relies on formulating the problem such that constraints of the original optimization problem and the structural analysis equations (imposed as equality constraints) form the constraint set for the problem. This
215 way the procedure only requires the solution of a nonlinear optimization problem using simultaneously the conventional sizing variables and the displacements as unknown variables. This approach, which is well suited for nonlinear structural problems, often results in a highly nonlinear problem that can become difficult to solve for large structures with many design variables. A more recent approach to the design with geometric nonlinearities is proposed by Buhl et al. (2000) for two-dimensional continuum structures. In order to reduce the computational effort associated with the optimization of a structure with nonlinear response, a displacement-based optimization procedure, which was originally implemented for optimization of trusses with linear response (Missoum et al. 1998), is proposed. Rather than solving the structural analysis equations repetitively inside a design variable optimization loop, the proposed method relies on solving for the optimal areas for specified displacements. The displacements specified for the inner loop are varied in an outer optimization loop until an optimal displacement field is determined. The optimization is no longer performed in the classical sizing-variable space, but in the displacement space as a two-level nested problem. 1. An inner problem in which the cross-sectional areas are evaluated for a given displacement field so that the structural weight is minimized. 2. An outer problem in which the optimal displacement field is searched in order to satisfy the constraints of the optimization problem. In this method, the design variables of the problem are, in some sense, the displacements, and the crosssectional areas are computed in the inner problem via a simple optimization problem. In the case of truss and membrane structures, the inner problem is a linear programming problem. The method has proven to be extremely efficient for truss structures that exhibit linear response, substantially reducing the computational time to reach an optimum solution (Missoum et al. 1998). The purpose of this article is to demonstrate the use of the displacement-based optimization for the solution of structural optimization problems with geometric and material nonlinearities. A description of the problem formulation will be presented and three test examples of plane and space trusses will be used to validate the method.
2 Description Before formulating the displacement-based method for nonlinear cases, a brief description of the procedure for the linear response is provided. As mentioned in the introduction, the method is based on two different levels referred to as the inner and the outer problems.
2.1 Definition of the inner problem The objective of the inner problem is to find the distribution of cross sectional areas for a given displacement field. This is done by solving an optimization problem that minimizes the structural weight W while satisfying the governing equilibrium equations. Using the standard finite element formulation, the equilibrium equations are given by K[a]u = f ,
(1)
which can be rewritten in the following form: T[u]a = f ,
(2)
where u is the displacement vector, a the vector of the m cross-sectional areas and f the vector of applied forces; K is the stiffness matrix of dimensions (n, n) where n is the number of degrees of freedom of the structure. The matrix T (n, m) in (2) is defined in terms of the truss geometry, material properties, and the nodal displacements. For specified nodal displacements, (2) is a set of linear equations in terms of the unknown area variable vector a. The linearity of (2) comes from the expression of the elementary stiffness matrices, which are simple linear functions of the areas. Details of the construction of the T matrix are given in the paper by Missoum et al. 1998). The inner problem can then be expressed for m members in the following form: m min W (a) = ρi ai , (3) i=1 T[u]a = f , where ρ is the material density and i is the ith member length. For a specified displacement field, this is clearly a linear problem in terms of the cross-sectional areas, which can be solved by linear programming. Lower and upper bounds on cross-sectional areas can easily be incorporated into the above formulation. Moreover, as pointed out by Missoum et al. 1998), the lower bounds of the areas can be set to zero exactly hence allowing to searching new topologies (Ringertz 1984; Sankaranarayanan et al. 1994; McKeown 1998). Opposite is the case with finite element analysis based optimizations where the member areas can only be reduced down to very small values to avoid singularities in the stiffness matrix. However, for large problems, the presence of very small element stiffnesses can lead to numerical difficulties. 2.2 Definition of the outer problem The outer problem is designed to search for the optimal displacement field corresponding to the optimal structural design. Although the designs obtained at the end of
216 each cycle of the inner loop are optimal, they are optimal for the specified displacements only. There is also an optimal weight corresponding to the specified displacements. This weight is an implicit function of the displacements, and can be reduced further by searching the displacement space. The outer level is formulated as an optimization problem in terms of the displacements in which the weight is minimized subjected to displacement and/or stress constraints min W (u) , gi (u) ≤ 0
i = 1, . . . , m .
(4)
Such a problem can be solved by any classical optimization method. However, as shown by Missoum et al. (1998) for the linear case, a derivative-based approach is preferred because of the availability of the gradient information. Indeed, the dual variables λ associated with the equality constraints of the inner problem provide directly the sensitivity of the weight with respect to the applied force f (Haftka and G¨ urdal 1992). That is, we have ∂W =λ. ∂f
becomes the definition of the tangent stiffness matrix KT . Finally, we have ∂W = KT λ . ∂u
(7)
The derivatives of the implicit optimal weight with respect to the displacements can then be computed without requiring finite differences. Therefore, the weight may be linearized and (4) can be solved by a sequential linear programming algorithm or any gradient-based method. The flow chart for the complete optimization scheme is depicted in Fig. 1.
2.3 Treatment of geometric nonlinearities For problems with geometric nonlinearities, the components of the FEM stiffness matrix are no longer simple functions of the member characteristics, but are also dependent on the displacements. The equilibrium equations in this case are written in the following form:
(5) K[a, u]u = f .
In order to find the derivatives of the weight with respect to the displacements, a simple chain rule can be used ∂W ∂f ∂W ∂f = = λ. ∂u ∂u ∂f ∂u
(6)
In the linear case, the derivative of f with respect to u is the stiffness matrix. When considering nonlinearities, it
u0 Inner Problem Find the distribution of areas a for specified displacements ∂W = KT λ ∂u Outer Problem min W (u) gi (u) ≤ 0 i = 1, . . . , m → uk+1 Fig. 1 Displacement based optimization scheme
(8)
The above set of nonlinear equations is commonly solved for the unknown displacements by an iterative process like the Newton-Raphson or the arc length method. Such solution strategies are computationally more expensive than the solution of the set of linear equations (1). Therefore, embedding them inside a traditional optimization scheme can lead to an extremely expensive computational process. The advantage of the displacement-based optimization can be readily observed from the form of (8). For a specified displacement field, (8) can still be put into the form shown in (2), which constitutes a set of linear equations in terms of the cross-sectional areas. The only major difference from the linear case is the fact that the matrix equilibrium equations represented by (2) are based on deformed state rather than the original undeformed state. The equilibrium equations have to be updated with respect to two contributions: the changes in the member orientations, and strain nonlinearities. In order to demonstrate the form of the equilibrium equations for the inner level problem, the force equilibrium of a truss node is considered. Writing the equilibrium of node j attached to p elements, we have p
fij,k + fext,j,k = 0 ,
(9)
i=1
where fij is the force applied to node j by element i and fext,j are the external forces applied to node j. The subscript k indicates the coordinate axis x, y, or z.
217 As an example, we consider an element i connected to nodes 1 and 2. The coordinates and the displacements of those nodes are {(x1 , y1 ), (x2 , y2 )} and {(u1 , v1 ), (u2 , v2 )}, respectively (Fig. 2). Expressing the force in terms of the stress, we have
p i=1
Ei
(u) − 0 (x2 − x1 ) + (u2 − u1) ai = 0 . 0 (u)
(14)
where 0 is the original length of the ith element and (u) is the updated length (u) = X 2 + Y 2 ,
These equations show that, in a displacement-based approach, the changes required to incorporate nonlinearities do not require any extra effort. Indeed, the quantities appearing in (11)–(14) are functions of the displacements and can easily be calculated despite their nonlinearity. This diverges from the classical optimization scheme where the linear analysis has to be replaced by nonlinear analysis in order to evaluate the displacements. In that case, the nonlinearities introduced in (11)–(14) are taken into account using an incremental update of the tangent stiffness matrix. At the end of an optimization process with structural nonlinearities, the optimal displacement field (u∗ ) and the corresponding set of cross-sectional areas (a∗ ) should satisfy the following system:
where
K[a∗ , u∗ ]u∗ = f .
fij = σij ai ,
(10)
where σij is the stress in element i connected to node j and ai is the cross-sectional area of the ith member. The stress can be expressed in terms of the nonlinear strain σij = Ei ij = Ei
(u) − 0 , 0
X = (x2 − x1 + u2 − u1 ) ,
(11)
Y = (y2 − y1 + v2 − v1 ) . (12)
(15)
The proposed method, which evolves in the displacement space, should then reach the exact solution of that problem without significant additional computational effort compared to that of a linear analysis based optimization.
2.4 Treatment of material nonlinearities
Fig. 2 Bar element in deformed and undeformed state
Projecting the forces on the axes, the updated direction cosines are evaluated by considering the displaced nodal positions and the updated length. We have for instance cos θi =
x2 − x1 + u2 − u1 , (u)
sin θi =
y2 − y1 + v2 − v1 . (u)
(13)
The equilibrium equations (9) for node j, projected on the x-axis, can now be written in terms of the displacements
Assuming the availability of a stress-strain (σ, ) law for a given material, the inclusion of material nonlinearities in the displacement-based process is quite straightforward as well. Because the displacements are specified at the outer level, the strains of the bar elements, and hence the stresses, can be easily computed based on the knowledge of the stress-strain relation. That is, the derivation of the equilibrium equations (9) and the construction of the inner problem in the case of material nonlinearities is a minor effort compared to the linear case. Moreover, the use of (7) to obtain the derivative of the weight with respect to all the displacement-components still holds. The only modification required is to update the tangent stiffness matrix with the tangent moduli ET = ∂σ ∂ available for each element from the stress-strain relation. Since the integration of material nonlinearities in the displacement-based optimization only requires the calculation of stresses directly from the stress-strain law, they can be easily combined with geometric nonlinearities. In order to validate the method described above, three test examples have been solved to show its capability in handling nonlinear optimization problems. The test problems used are three truss structures. The first two examples exhibit geometric nonlinearities only. The last example will demonstrate the use of the displacementbased method for combined geometric and material nonlinearities.
218 3 Test example results 3.1 Two-bar truss The first example is a symmetrical two bar truss, which is a single degree-of-freedom structure. Its geometry and dimensions are depicted in Fig. 3. A vertical force f is applied to the top node of the truss. For this problem, a simple expression for the geometrically nonlinear force-deflection relation is easy to obtain, and the following approximate solution is given by Christfield (1991): f≈
Ea 2 2h − 3hD2 + D3 , L30
The minimization of the weight is considered here without any other constraints. The displacement-based method (DBM) converges in five iterations, and the optimal solution corresponds to a geometry for which the applied load becomes a critical load. That is, the crosssectional area of the bars is reduced until the structure exhibits a snap-through for the applied load. The corresponding deflection is shown in Table 1, and agrees well with the analytical solution. The optimal area is 0.790 (dimensionless) and the corresponding force-deflection curve reaches its maximum at point (4.227, 90) (Fig. 4). The convergence of the method to the solution presented above is justified by the fact that any lighter design would not be able to withstand the applied load thus violating the equilibrium equations of the inner problem.
(16)
where L0 is the initial length and D is the deflection.
3.2 Space dome structure The structure considered is shown in Fig. 5. It is subjected to a 2000-pound point load on node 1 in the vertical downward (z) direction. All 30 cross-sectional areas have a lower bound of 0.1 in2 . The weight of the structure is to be minimized with no constraints other than the member area lower bounds.
Fig. 3 Two-bar truss
Table 1 Results obtained for the two bar truss with the displacement-based method. Comparison to the analytical solution Final Area (dimensionless)
Limit deflection
Analytical limit defl.
0.790
4.227
4.226
Fig. 5 Dome structure. Top view
Table 2 Results obtained for the space dome structure. Comparison to a literature result
Fig. 4 Force-deflection curve for the initial and final design of the two-bar truss
Areas (in2 )
Khot and Kamat (1985)
Displacement based method
1,2,3,4,5,6 7,8,9,10,11,12 13,16,19,22,25,28 Final weight (lb)
1.6926 1.3754 0.2693 766.19
1.6904 1.3833 0.2674 766.72
219 Khot and Kamat (1985) performed the optimization based on an optimality criterion by assuming that the optimal structure should have a uniform nonlinear strain energy density distribution. Doing so, the final weight obtained in that work was 766.19 lb. For the displacement-based approach, due to the symmetry of the problem, only three displacements are used as variables: node 1 vertical displacement and node 3 vertical and radial displacement (Fig. 5). The design obtained is presented in Table 2, which agrees well with the results provided by Khot and Kamat (1985). Again, the design is such that the applied load has become a limit load. This can be shown on the force-deflection curve (Fig. 6) where the point (10, 2000) is critical. The CPU time required to obtain the solution with the displacement-based method is 3.6 s on a PC Pentium 150 MHz. This value, which is small in an absolute sense, cannot be compared to the article result since the authors do not provide such information. However, the CPU time required for the full optimization in DBM is comparable to computational time required for a single nonlinear analysis.
displacement-based method uses the eight free degrees of freedom as variables. For this optimization problem, three different cases will be considered: 1. structural response with geometric nonlinearities only 2. structural response with material nonlinearities only 3. structural response combining both material and geometric nonlinearities
Fig. 7 Ten-bar truss
Table 3 Results obtained for the ten bar truss with geometric nonlinearities
Fig. 6 Force-deflection curve for the dome structure
3.3 Ten-bar truss The third example used to illustrate the methodology is the popular ten-bar truss depicted in Fig. 7. It is subjected to two forces of 107 lbs on nodes 2 and 4. The Young’s Modulus of the material is 107 psi. The optimization problem considered is the minimization of the weight with two displacement constraints of 100 inches in the vertical direction on node 1 and 2. The ten member cross sectional areas have a lower bound of 0.1 in2 . The
Areas (in2 )
Classical Opti.
DBM
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Weight (lb) CPU time (sec)
56.160 0.368 42.118 21.545 0.100 0.100 22.883 39.858 38.622 0.486 9519.53 12.3
56.160 0.368 42.118 21.545 0.100 0.100 22.883 39.850 38.622 0.486 9519.53 1.0
3.3.1 Geometric nonlinearities The results of the DBM optimization including geometric nonlinearities are presented in Table 3. They are compared to the classical finite element based optimization. The optimal solutions obtained in both cases are identical with a weight of 9519.53 pounds and both displacement constraints active. The major difference lies in the CPU time. The results were obtained in 12.3 s
220 using the classical method compared to 1.0 s using the DBM.
3.3.2 Material nonlinearities The material used is assumed to follow a bilinear stressstrain relation (linear hardening) shown in Fig. 8. The modulus of the hardening part is set to Eh = 5.0 × 106 psi and the yield stress value is σy = 25 000 psi. Based on material nonlinearities only, the solution obtained with the DBM is summarized in Table 4 and matches the optimal solution using the classical approach. In this case, the final weight is 18 840.1 lb and both displacement constraints are active. The CPU time required for the classical method is 12.2 s compared to the 0.7 s for the DBM.
in Table 5 show that the classical and displacementbased method reach the same solution with a final weight of 17 791.2 lb and both displacement constraints active. Here the computational times are 11.2 s and 0.8 s, respectively, for the classical and displacement-based approaches.
Table 5 Results obtained for the ten-bar truss with geometric and material nonlinearities Areas (in2 )
Classical Opti.
DBM
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Weight (lb) CPU time (sec)
104.746 0.407 79.036 39.578 0.100 0.100 44.176 74.325 72.030 0.550 17 791.2 11.2
104.746 0.407 79.036 39.578 0.100 0.100 44.176 74.325 72.030 0.550 17 791.2 0.8
4 Discussion
Fig. 8 Stress-strain curve of the material used for the ten bar truss
Table 4 Results obtained for the ten bar truss with material nonlinearities Areas (in2 )
Classical Opti.
DBM
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 Weight (lb) CPU time (sec)
111.696 0.312 85.155 50.373 0.100 0.100 38.411 78.708 77.346 0.412 18 840.1 12.2
111.696 0.312 85.155 50.373 0.100 0.100 38.411 78.708 77.346 0.412 18 840.1 0.7
3.3.3 Material and geometric nonlinearities In this example, the problem is solved using the material and geometric effects simultaneously. The results
4.1 Topology of the displacement space One of the first questions that arise when describing the displacement-based approach is concerned with the possibility of having a nonfeasible displacement field generated at the outer level optimization. That is, there may not be any combination of cross-sectional areas capable of generating the specified displacements, and hence the inner problem may not have a feasible solution. In the linear case (Missoum et al. 1998; McKeown 1989), a subset of the infeasible region of the displacement space was defined by those points for which the quantity f T · u is negative implying a negative value for the strain energy within the structure, which is impossible. At the boundary of that infeasible region, the weight becomes infinite, forcing the optimization process to remain within the feasible region. In the nonlinear case, the description of the search space becomes somewhat more complicated. The presence of possible points of instability has to be taken into account as well. These points in the displacement space are such that the applied load becomes a limit load or that a bifurcation point is reached. In either case, the process cannot progress any further. The two-bar truss and the dome are two examples of such a behaviour. For those two structures, the limit points correspond clearly to solutions in the displacement space where the weight
221 cannot be minimized further without a violation of the equilibrium equations. These remarks raise the question of the importance of the starting displacement field for the optimization process. For all the experiments made so far, the initial displacement fields were obtained from either a linear or nonlinear finite element analysis of a meaningful truss design, which enables us to start the optimization process in the stable and feasible region. 4.2 Computational time As for the linear case, the reduction in computational time when compared to a classical finite element based optimization has two main sources. 1. The replacement of a nonlinear FE analysis by a simpler and cheaper linear programming problem. 2. The availability of the derivatives of the objective function with respect to the displacements using the dual variables of the inner problem. This avoids the burden of repeated finite differences that are needed to compute the derivatives. These features become very attractive for nonlinear problems for which a single analysis can reach several minutes even for small truss structures. Moreover, nonlinear algorithms often require a tuning of parameters (such as number of substeps, convergence within a substep, etc.), and might fail to perform an analysis accurately. This can lead to difficulties if the analysis is embedded within an optimization loop.
5 Conclusions An extension of the displacement-based method to problems including geometric and material nonlinearities has been presented. The advantage of the procedure relies on the fact that the displacements are specified at an outer level. This way, any modification of the stiffness matrix with respect to the displacements is easy to implement. The method avoids the use of repeated nonlinear finite element analyses, which are very often computationally expensive, and hence it is fast and accurate. Moreover, the use of the dual variables of the inner problem allows obtaining the exact sensitivities of the weight with
respect to the displacements, which in turn reduces substantially the computational time by avoiding the use of repeated finite differences. A proof of the efficiency has been provided on three test examples including geometric and material nonlinearities. More research is warranted to show the efficiency of the method for larger structures and its application to beam and plate problems. Acknowledgements This research was partially funded by the Design Optimization Vehicle Safety Research Department of the Ford Motor Company under University Research Program. Dr. Ren-Jye Yang was the project monitor.
References Buhl, T.; Pedersen, C.B.W.; Sigmund, O. 2000: Stiffness design of geometrically nonlinear structures using topology optimization. Struct. Multidisc. Optim. 19, 93–104 Christfield, M.A. 1991: Nonlinear finite element analysis of solids and structures. Chichester, England: Wiley Haftka, R.T.; Kamat, M.P. 1985: Simultaneous nonlinear structural analysis and design. Comp. Mech. 14 , 409–416 Haftka, R.T.; G¨ urdal, Z. 1992: Elements of structural optimization. Boston, MA: Kluwer Khot, N.S.; Kamat, M.P. 1985: Minimum weight design of truss structures with geometric nonlinear behavior, AIAA Journal.23, 139–144 McKeown, J.J. 1989: The design of optimal trusses via sequences of optimal fixed displacement structures. Eng. Opt. 14, 159–178 McKeown, J.J. 1998: Growing optimal pin-jointed frames. Struct. Optim. 15, 92–100 Missoum, S.; G¨ urdal, Z.; Hernandez, P.; Guillot, J. 1998: A displacement based optimization for truss structures subjected to static and dynamic constraints. Proc. 7th AIAA/NASA/ISSMO Symp. on Multidisciplinary and Structural Optimization (held in St. Louis, MO), pp. 681–690 Ringertz, U.T. 1984: On topology optimization of trusses. Eng. Opt. 9, 209–218 Ringertz, U.T. 1988: Optimization of structures with nonlinear response structural optimization. Eng. Opt., 123–156 Sankaranarayanan, S.; Haftka, R.T.; and Kapania, R.K. 1994: Truss topology optimization with simultaneous analysis and design. AIAA J. 32, 420–424
