Contact shape optimization: a bilevel programming approach

We consider the problem of shape optimization of nonlinear elastic solids in contact. The equilibrium of the solid is defined by a constrained minimization problem, where the body energy functional is the objective and the constraints impose the nonpenetration condition. Then the optimization problem can be formulated in terms of a bilevel mathematical program. We describe new optimality conditions for bilevel programming and construct an algorithm to solve these conditions based on Herskovits’ feasible direction interior point method. With this approach we simultaneously carry out shape optimization and nonlinear contact analysis. That is, the present method is a “one shot” technique. We describe some numerical examples solved in a very efficient way. Received July 27, 1999     

Topology optimization of softening structures under displacement constraints as an MPEC

This paper presents a mathematical programming based technique for the minimum weight (volume) topology optimization of truss-like structures such that strain softening material properties, that can lead to severe physical instability, can be accommodated. In addition, satisfaction of such serviceability criteria as limited displacements at some specific points is ensured. The problem is formulated in terms of truss member cross-sectional areas. This leads to a challenging nonconvex and nonsmooth optimization problem, known as a mathematical program with equilibrium constraints (MPEC). A two-step optimization algorithm is proposed to overcome the problems typically associated with nondefiniteness of some key matrices and at the same time nondifferentiability of the mathematical system. Each step involves updating the ground structure and solving the MPEC using a penalized nonlinear programming (NLP) approach. Some numerical examples are provided to illustrate application, robustness and efficiency of the proposed scheme. The safety and integrity of the designed topologically optimal structures are validated using appropriate stepwise holonomic elastoplastic analyses.     

Stochastic bilevel programming in structural optimization

We consider the mathematical modelling and solution of robust and cost-optimizing structural (topology) design problems. The setting is the optimal design of a linear-elastic structure, for example a truss topology, under unilateral frictionless contact, and under uncertainty in the data describing the load conditions, the material properties, and the rigid foundation. The resulting stochastic bilevel optimization model finds a structural design that responds the best to the given probability distribution in the data. This model is of special interest when a structural failure will lead to a reconstruction cost, rather than loss of life. For the mathematical model, we provide results on the existence of optimal solutions which allow for zero lower design bounds. We establish that the optimal solution is continuous in the lower design bounds, a result which validates the use of small but positive values of them, and for such bounds we also establish the locally Lipschitz continuity and directional differentiability of the implicit upper-level objective function. We also provide a heuristic algorithm for the solution of the problem, which makes use of its differentiability properties and parallelization strategies across the scenarios. A small set of numerical experiments illustrates the behaviour of the stochastic solution compared to an average-case deterministic one, establishing an increased robustness. Received December 22, 1999     

Constraint programming approach to a bilevel scheduling problem

Bilevel optimization problems involve two decision makers who make their choices sequentially, either one according to its own objective function. Many problems arising in economy and management science can be modeled as bilevel optimization problems. Several special cases of bilevel problem have been studied in the literature, e.g., linear bilevel problems. However, up to now, very little is known about solution techniques of discrete bilevel problems. In this paper we show that constraint programming can be used to model and solve such problems. We demonstrate our first results on a simple bilevel scheduling problem.     

Topology optimization with finite-life fatigue constraints

This work investigates efficient topology optimization for finite-life high-cycle fatigue damage using a density approach and analytical gradients. To restrict the minimum mass problem to withstand a prescribed finite accumulated damage, constraints are formulated using Palmgren-Miner’s linear damage hypothesis, S-N curves, and the Sines fatigue criterion. Utilizing aggregation functions and the accumulative nature of Palmgren-Miner’s rule, an adjoint formulation is applied where the amount of adjoint problems that must be solved is independent of the amount of cycles in the load spectrum. Consequently, large load histories can be included directly in the optimization with minimal additional computational costs. The method is currently limited to proportional loading conditions and linear elastic material behavior and a quasi-static structural analysis, but can be applied to various equivalent stress-based fatigue criteria. Optimized designs are presented for benchmark examples and compared to stress optimized designs for static loads.     

Topology optimization of trusses with local stability constraints and multiple loading conditions—a heuristic approach

In truss topology optimization against buckling constraints, the extension from considering a single load case to include multiple loading conditions remains an unsolved problem in the ground structure approach. The present paper suggests a heuristic method attempting to take the multiple load situation into account. A method by Pedersen (1993, 1994) considering only single loading conditions is generalized to include multiple load cases. Based on the ground structure approach the algorithm allows for variable ground structures allowing for, for instance, geometrical restrictions such as concave or even disconnected design domains (Smith 1995b).     

一类非线性两层规划问题的递阶优化解法

提出一种求解一类非线性两层规划问题的新方法．通过引入解耦向量将非线性两层规划问题分解为独立且易于求解的子问题，利用两级递阶结构第1级求解若干优化的子问题，而在第2级利用第1级求解的结果调整解耦向量．所提出的方法借助于分解一协调原理并按迭代方式最终求得问题的最优解．对于含整数的规划问题，通过连续化处理后也可按该方法方便地求解．算例表明所提出的算法是简便而有效的．     

Design optimization with static and dynamic displacement constraints

A design optimization procedure using a sequential linear programming technique is proposed in this paper to design minimum weight structures subjected to frequency response and static displacement constraints. The merit of the proposed approach is that the reanalyses of the static and dynamic responses, as well as the computations of the static and dynamic sensitivity data, are performed in a reduced approximate model. A significant saving of computer time for large scale structures is expected. Two numerical examples show good results of this method.     

Topology optimization of continuum structures with material failure constraints

This work presents an efficient strategy for dealing with topology optimization associated with the problem of mass minimization under material failure constraints. Although this problem characterizes one of the oldest mechanical requirements in structural design, only a few works dealing with this subject are found in the literature. Several reasons explain this situation, among them the numerical difficulties introduced by the usually large number of stress constraints. The original formulation of the topological problem (existence/non-existence of material) is partially relaxed by following the SIMP (Solid Isotropic Microstructure with Penalization) approach and using a continuous density field as the design variable. The finite element approximation is used to solve the equilibrium problem, as well as to control through nodal parameters. The formulation accepts any failure criterion written in terms of stress and/or strain invariants. The whole minimization problem is solved by combining an augmented Lagrangian technique for the stress constraints and a trust-region box-type algorithm for dealing with side constraints (0<_min1) . Numerical results show the efficiency of the proposed approach in terms of computational costs as well as satisfaction of material failure constraints. It is also possible to see that the final designs define quite different shapes from the ones obtained in classical compliance problems.     

最小位移类桁架连续体拓扑设计优化

为将拓扑优化中的柔度最小化问题拓展到一般位移最小化问题,用有限元划分设计域,采用类桁架连续体材料模型,并假设杆件在设计域内连续分布.将杆件在节点位置的密度和方向作为设计变量,将指定位置和方向的位移作为目标函数,采用基于目标函数梯度的优化准则法,通过优化杆件的连续分布场形成拓扑优化的类桁架连续体.该方法可结合结构力学的基本概念,选择部分杆件形成拓扑优化刚架.     

Topology optimization of fluid channels with flow rate equality constraints

This note presents topology optimization of fluid channels with flow rate equality constraints. The equality constraints on the specified boundaries are implemented using the lumped Lagrange multiplier method. The quadratic penalty term and cut-off sensitivity are used to maintain the stability of optimization.     

Topology optimization for minimum weight with compliance and stress constraints

The paper deals with a formulation for the topology optimization of elastic structures that aims at minimizing the structural weight subject to compliance and local stress constraints. The global constraint provides the expected stiffness to the optimal design while a selected set of local enforcements require feasibility with respect to the assigned strength of material. The Drucker?CPrager failure criterion is implemented to handle materials with either equal or unequal behavior in tension and compression. A suitable relaxation of the equivalent stress measure is implemented to overcome the difficulties related to the singularity problem. Numerical examples are presented to discuss the features of the achieved optimal designs along with performances of the adopted procedure. Comparisons with pure compliance?Cbased or pure stress?Cbased strategies are also provided to point out differences arising in the optimal design with respect to conventional approaches, depending on the assumed material behavior.     

A neural network approach for solving nonlinear bilevel programming problem

A neural network model is presented for solving nonlinear bilevel programming problem, which is a NP-hard problem. The proposed neural network is proved to be Lyapunov stable and capable of generating approximal optimal solution to the nonlinear bilevel programming problem. The asymptotic properties of the neural network are analyzed and the condition for asymptotic stability, solution feasibility and solution optimality are derived. The transient behavior of the neural network is simulated and the validity of the network is verified with numerical examples.     

A neural network approach for solving linear bilevel programming problem

A novel neural network approach is proposed for solving linear bilevel programming problem. The proposed neural network is proved to be Lyapunov stable and capable of generating optimal solution to the linear bilevel programming problem. The numerical result shows that the neural network approach is feasible and efficient.     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

Topology optimization of cellular materials with periodic microstructure under stress constraints

Material design is a critical development area for industries dealing with lightweight construction. Trying to respond to these industrial needs topology optimization has been extended from structural optimization to the design of material microstructures to improve overall structural performance. Traditional formulations based on compliance and volume control result in stiffness-oriented optimal designs. However, strength-oriented designs are crucial in engineering practice. Topology optimization with stress control has been applied mainly to (macro) structures, but here it is applied to material microstructure design. Here, in the context of density-based topology optimization, well-established techniques and analyses are used to address known difficulties of stress control in optimization problems. A convergence analysis is performed and a density filtering technique is used to minimize the risk of results inaccuracy due to coarser finite element meshes associated with highly non-linear stress behavior. A stress-constraint relaxation technique (qp-approach) is applied to overcome the singularity phenomenon. Parallel computing is used to minimize the impact of the local nature of the stress constraints and the finite difference design sensitivities on the overall computational cost of the problem. Finally, several examples test the developed model showing its inherent difficulties.

     

Topology optimization of continuum structures with local and global stress constraints

Topology structural optimization problems have been usually stated in terms of a maximum stiffness (minimum compliance) approach. The objective of this type of approach is to distribute a given amount of material in a certain domain, so that the stiffness of the resulting structure is maximized (that is, the compliance, or energy of deformation, is minimized) for a given load case. Thus, the material mass is restricted to a predefined percentage of the maximum possible mass, while no stress or displacement constraints are taken into account. This paper presents a different strategy to deal with topology optimization: a minimum weight with stress constraints Finite Element formulation for the topology optimization of continuum structures. We propose two different approaches in order to take into account stress constraints in the optimization formulation. The local approach of the stress constraints imposes stress constraints at predefined points of the domain (i.e. at the central point of each element). On the contrary, the global approach only imposes one global constraint that gathers the effect of all the local constraints by means of a certain so-called aggregation function. Finally, some application examples are solved with both formulations in order to compare the obtained solutions.     

An interval programming approach for the bilevel linear programming problem under fuzzy random environments

In this paper, we address a class of bilevel linear programming problems with fuzzy random variable coefficients in objective functions. To deal with such problems, we apply an interval programming approach based on the $\alpha $ -level set to construct a pair of bilevel mathematical programming models called the best and worst optimal models. Through expectation optimization model, the best and worst optimal problems are transformed into the deterministic problems. By means of the Kth best algorithm, we obtain the best and worst optimal solutions as well as the corresponding range of the objective function values. In this way, more information can be provided to the decision makers under fuzzy random circumstances. Finally, experiments on two examples are carried out, and the comparisons with two existing approaches are made. The results indicate the proposed approaches can get not only the best optimal solution (ideal solution) but also the worst optimal solution, and is more reasonable than the existing approaches which can only get a single solution (ideal solution).     

Constrained optimization of structures with displacement constraints under various loading conditions

Optimization problems often involve constraints and restrictions which must be considered in order to obtain an optimum result and the resultant solution should not deviate from any of the imposed constraints. These constraints and restrictions are imposed either on the design variables or on the algebraic relations between them. Constraints of allowable stress, minimum size and buckling of members in the absence of allowable displacement constraint are the most important factors in optimization of the cross-sectional area of structural elements. When the allowable displacement constraint is included in the problem as a determinant parameter, since the specifications of most of elements affect the displacement rate, the way of imposing and considering this constraint requires special care. In this research the way of simultaneous imposition of multi displacement constraints for optimum design of truss structures in several load cases is described. In this method various constraints for different load cases are divided into active and passive constraints. The mathematical formulation is based on the classical method of Lagrange Multipliers. Overall, this simple method can be employed along with other constraints such as buckling, allowable stress and minimum size of members for imposing the displacement constraint in various load cases.     

Topology optimization of a cantilevered piezoelectric energy harvester using stress norm constraints

Vibrational piezoelectric energy harvesters are devices which convert ambient vibrational energy into electric energy. Here we focus on the common cantilever type in which an elastic beam is sandwiched between two piezoelectric plates. In order to maximize the electric power for a given sinusoidal vibrational excitation, we perform topology optimization of the elastic beam and tip mass by means of the SIMP approach, leaving the piezoelectric plates solid. We are interested in the first and especially second resonance mode. Homogenizing the piezoelectric strain distribution is a common indirect approach increasing the electric performance. The large design space of the topology optimization approach and the linear physical model also allows the maximization of electric performance by maximizing peak bending, resulting in practically infeasible designs. To avoid such problems, we formulate dynamic piezoelectric stress constraints. The obtained result is based on a mechanism which differs significantly from the common designs reported in literature.