Buckling Constraints Research Articles

Topology optimization of stability‐constrained structures with simple/multiple eigenvalues

AbstractThis work focuses on topology optimization formulations with linear buckling constraints wherein eigenvalues of arbitrary multiplicities can be canonically considered. The non‐differentiability of multiple eigenvalues is addressed by a mean value function which is a symmetric polynomial of the repeated eigenvalues in each cluster. This construction offers accurate control over each cluster of eigenvalues as compared to the aggregation functions such as ‐norm and Kreisselmeier–Steinhauser (K–S) function where only approximate maximum/minimum value is available. This also avoids the two‐loop optimization procedure required by the use of directional derivatives (Seyranian et al. Struct Optim. 1994;8(4):207‐227.). The spurious buckling modes issue is handled by two approaches—one with different interpolations on the initial stiffness and geometric stiffness and another with a pseudo‐mass matrix. Using the pseudo‐mass matrix, two new optimization formulations are proposed for incorporating buckling constraints together with the standard approach employing initial stiffness and geometric stiffness as two ingredients within generalized eigenvalue frameworks. Numerical results show that all three formulations can help to improve the stability of the optimized design. In addition, post‐nonlinear stability analysis on the optimized designs reveals that a higher linear buckling threshold might not lead to a higher nonlinear critical load, especially in cases when the pre‐critical response is nonlinear.

Lattice structure design optimization under localized linear buckling constraints

An optimization method for the design of multi-lattice structures satisfying local buckling constraints is proposed in this study. In order to resolve the highly nonlinear large-scale optimization problem, a consecutive numerical approach is devised: macro-field optimization and microstructure embedding. The macro-field optimization finds an optimal elasticity tensor distribution among all feasible elastic continua based on a concept of free material optimization (FMO), which largely extends the design space compared with density-based topology optimization. After this, by an inverse design approach that approximates the elasticity tensor under local buckling constraint, an appropriate lattice structure is able to be embedded within each macro-element, where the local stress tensors are considered in particular to produce more reasonable microstructures. During the process, a machine learning approach is also devised to significantly reduce the number of material/lattice types and thus the overall computational costs. Ultimately, a lattice structure under requirements of overall stiffness and local buckling resistance is produced, as demonstrated via numerical examples.

Efficient buckling constrained topology optimization using reduced order modeling

We present an efficient computational approach to continuum topology optimization with linearized buckling constraints, using Reduced Order Models (ROM). A reanalysis technique is employed to generate basis vectors that subsequently are used to significantly reduce the size of the generalized eigenvalue problems. We demonstrate the efficacy of this approach by optimizing for stiffness with buckling constraints and show results for several test cases. Based on our findings, we conclude that the ROM can potentially save significant computational effort without compromising the quality of the results.

Finite strain topology optimization with nonlinear stability constraints

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm — the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching, and multiple bifurcations are also demonstrated.

Density-based topology optimization integrated with genetic algorithm for optimizing formability and bending stiffness of 3D printed CFRP core sandwich sheets

To expand the application ranges of sandwich sheets from conventional 2D flat panel types to 3D complex shapes and simultaneously improve the specific bending stiffness , a novel topology optimization strategy that can optimize the bending stiffness while maintaining good formability was proposed. In the proposed approach, the density-based topology optimization was integrated with the multi-stage genetic algorithm (GA) to optimize the repeatable unit cell of the core structure of sandwich sheets. Two optimization schemes were adopted, in which one optimizes the formability and the other one optimizes the bending stiffness while fulfilling potential failure constraints. The failure constraints on core shear failure and face buckling were theoretically deduced to mathematically formulate the topology optimization problem, which was solved by the adaptive multi-stage GA to increase the possibility of generating physically meaningful and additively manufacturable topologies. For the experimental evaluations of mechanical properties and formability, the final optimal topologies under volume fraction constraints of 50% and 62.5% were additively manufactured using carbon fibre reinforced nylon. Comparing the sandwich topologies obtained by two optimization schemes, the bending stiffness of sandwich topologies with the core density of 50% and 62.5% are improved by 41.58% and 41.49%, while the energy absorption capabilities are improved by 13.60% and 29.40% respectively. L-bending and draw-bending tests indicate the improved formability of topologically optimized sandwich sheets. The proposed approach is capable of designing formable sandwich sheets with improved bending stiffness, which is expected to expand the application envelope of sandwich sheets with better mechanical properties. • Topology optimization of sandwich sheets that can not only be bent without failure but also have optimal bending stiffness. • Integration of density-based topology optimization with multi-stage genetic algorithm. • Theoretical core shear failure and face buckling constraints on sandwich sheets during bending. • Improved bending stiffness and formability of topologically optimized sandwich sheets with 3D printed CFRP cores.

Discrete Material and Thickness Optimization of sandwich structures

In this paper Discrete Material and Thickness Optimization (DMTO) is used to optimize sandwich composite structures subject to both displacement and linear buckling constraints. Using a new thickness formulation where density design variables scale ply thicknesses rather than constitutive properties, it is possible to size both core and face sheet plies simultaneously. This makes it possible to have different ply thicknesses for core and face sheet layers while also covering ply-drops. Furthermore, separating core and face sheets allows enforcing a symmetric lay-up which can be important to avoid warping during curing. The approach is demonstrated in three numerical examples of increasing complexity.

Truss topology optimization considering local buckling constraints and restrictions on intersection and overlap of bar members

This paper illustrates the application of a two-level approximation method for truss topology optimization with local member buckling constraints and restrictions on member intersections and overlaps. Previously developed for truss topology optimization with stress and displacement constraints, that method is achieved by starting from an initial ground structure, and, combined with genetic algorithm (GA), it can handle both discrete and continuous variables, which denote the existence and cross-sectional areas of bar members respectively in the ground structure. In this work, this method is improved and extended to consider member buckling constraints and restrict intersection and overlap of members for truss topology optimization. The temporary deletion technique is adopted to temporarily remove buckling constraints when related bar members are deleted, and in order to avoid unstable designs, the validity check for truss topology configuration is conducted. By using GA to search in each possible design subset, the singularity encountered in buckling-constrained problems is remedied, and meanwhile, as the required structural analysis is replaced with explicit approximation functions in the process of executing GA, the computational cost is significantly saved. Moreover, for the consideration of restrictions on member intersecting and overlapping, the definition of such phenomena and mathematical expressions to recognize them are presented, and a new fitness function is developed to include such considerations. Numerical examples are presented to show the efficacy of the proposed techniques.

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

SummaryLinear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient‐based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level‐set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub‐problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.

Plate/shell topological optimization subjected to linear buckling constraints by adopting composite exponential filtering function

In this paper, a model of topology optimization with linear buckling constraints is established based on an independent and continuous mapping method to minimize the plate/shell structure weight. A composite exponential function (CEF) is selected as filtering functions for element weight, the element stiffness matrix and the element geometric stiffness matrix, which recognize the design variables, and to implement the changing process of design variables from “discrete” to “continuous” and back to “discrete”. The buckling constraints are approximated as explicit formulations based on the Taylor expansion and the filtering function. The optimization model is transformed to dual programming and solved by the dual sequence quadratic programming algorithm. Finally, three numerical examples with power function and CEF as filter function are analyzed and discussed to demonstrate the feasibility and efficiency of the proposed method.

Trim and Structural Optimization of Subsonic Transport Wings Using Nonconventional Aeroelastic Tailoring

Several minimum-mass aeroelastic optimization problems are solved to evaluate the effectiveness of a variety of novel tailoring schemes for subsonic transport wings. Aeroelastic strength and panel buckling constraints are imposed across several trimmed maneuver loads, in addition to flutter constraints. Tailoring with metallic thickness variations, functionally graded materials, composite laminates, tow steering within composite laminates, and distributed trailing-edge control effectors are all found to provide reductions in structural wing mass with varying degrees of success. The question as to whether this wing mass reduction will offset the increased manufacturing cost is left unresolved for each case.

Topological Optimization of Plate Subjected to Linear Buckling Constraints Based on Independent Continuous Mapping Method

In this paper, a model of topology optimization for the lightest plate structures with linear buckling constraints is constructed based on Independent, Continuous and Mapping (ICM) method. Exponential function is adopted as filtering function of the element weight, the element stiffness matrix and the element geometric stiffness matrix. Based on the Taylor expansion and the filtering function, the buckling constraints were approximately expressed as explicit functions. The optimization model was translated into a dual programming and solved by the sequence second-order programming. Two numerical examples show that this method is validity and efficient.

Thermo-structural optimization of integrated thermal protection panels with one-layer and two-layer corrugated cores based on simulated annealing algorithm

Toexplore weight saving potential capability, a multidisciplinary optimization procedure based on simulated annealing algorithm was proposed to unveil the minimum weight design for integrated thermal protection system subjected to in-service thermal and mechanical loads. The panel configurations with one-layer and two-layer corrugated cores are considered for comparison. Heat transfer and structural field analysis for each panel configuration were performed to obtain the temperature, buckling, stress and deflection responses for structural components of interest, which were then considered as critical constraints of the optimization problem. Sensitivity analysis was performed to disclose the effect of individual design variables on the thermo-structural extreme responses, and the designed thermal protection system performance and weight for the two configurations were discussed. The results demonstrated that the two-layer structure provides superior structural efficiency and performance to resist thermal buckling deformation in comparison with the one-layer panel. Its area-specific weight is reduced by more than 14---29 % with respect to the one-layer panel design, and 30---50 % weight efficient can be implemented at higher thermal buckling constraint levels, while keeping considerable temperature, stress and deflection margins.

Optimization of Blade Stiffened Composite Panel under Buckling and Strength Constraints

This paper deals with multiple constraints for dimension and stacking-sequence optimization of a blade-stiffened composite panel. In a previous study, a multiple objective genetic algorithm using a Kriging response surface with a buckling load constraint was the target. The present study focuses on dimension and stacking-sequence optimization with both a buckling load constraint and a fracture constraint. Multiple constraints complicate the process of selecting sampling analyses to improve the Kriging response surface. The proposed method resolves this problem using the most-critical-constraint approach. The new approach is applied to a blade stiffened composite panel and the approach is shown to be efficient.

Reliability optimization using probabilistic sufficiency factor and correction response surface

Reliability-based design optimization for low failure probability often requires millions of function analyses. Response surface approximation of the response functions (analysis response surface(ARS)) is often used to reduce the cost of failure probability calculations. Failure probabilities obtained from numerical sampling schemes are noisy and unsuitable for gradient-based optimization. To overcome this, response surfaces have been fitted to the failure probability of the designs (design response surface (DRS)) as a function of the design variables and used in optimization. Two shortcomings of the approach are that (i) the ARS fitting is extremely expensive for a large number of variables, especially for the high accuracy required to obtain very accurate reliability estimates, and (ii) DRS introduces fitting errors which affect the tails of the distributions which are significant for low failure probabilities. An approach to obtaining high-accuracy reliability estimates using the probabilistic sufficiency factor and correction response surface is investigated in this article. The method is demonstrated using a thin-walled box beam structure designed for minimum weight with failure probability constraints. The design is subjected to buckling, strength, and displacement constraints. Two methods of correcting low-fidelity analyses are compared for accuracy and efficiency. It is shown that correction to the response function is more accurate than the correction fitted to the probabilistic sufficiency factor.

On the design optimization of built-up stiffened steel beams with buckling and frequency constraints

The literature on the structural design optimization of steel-plate girders indicates a need for more refined research studies to obtain optimal designs by formulating and solving the design problem that combines structural sizing and shape parameters in one unified, constrained problem. For this purpose, the structural optimization design problem of stiffened steel-plate girders is formulated with specified loading conditions and constraints on strength and serviceability considerations including limits on fundamental frequency and buckling modes. The finite-element method-based model is used to define the objective function and the structural/geometric response functions, while the geometric domain elements are used to systematically perturb the structural shape during the search for an optimal shape of the structure. The mathematical statement of the gradient-based-design problem is solved for an optimal structural size and shape with buckling and frequency constraints in addition to the traditional strength constraints. The numerical results obtained are compared with results obtained from a less formal ad hoc design procedure, and some conclusions are drawn to emphasize the design benefits obtained from solving the design problem for optimal structural size and shape.

Optimal lay-up of hybrid composite beams, plates and shells using cellular genetic algorithm

Laminated composite structures find wide range of applications in many branches of technology. They are much suited for weight sensitive structures (like aircraft) where thinner and lighter members made of advanced fiber reinforced composite materials are used. The orientations of fiber direction in layers and number of layers and the thickness of the layers as well as material of composites play a major role in determining the strength and stiffness. Thus the basic design problem is to determine the optimum stacking sequence in terms of laminate thickness, material and fiber orientation. In this paper, a new optimization technique called Cellular Automata (CA) has been combined with Genetic Algorithm (GA) to develop a different search and optimization algorithm, known as Cellular Genetic Algorithm (CGA), which considers the laminate thickness, angle of fiber orientation and the fiber material as discrete variables. This CGA has been successfully applied to obtain the optimal fiber orientation, thickness and material lay-up for multi-layered composite hybrid beams plates and shells subjected to static buckling and dynamic constraints.

Minimum weight design of non‐linear elastic structures with multimodal buckling constraints

AbstractIt well known that multimodal instability is an event particularly relevant in structural optimization. Here, in the context of non‐linear stability theory, an exact method is developed for minimum weight design of elastic structures with multimodal buckling constraints. Given an initial design, the method generates a sequence of improved designs by determining a sequence of critical equilibrium points related to decreasing values of the structural weight. Multimodal buckling constraints are imposed without repeatedly solving an eigenvalue problem, and the difficulties related to the non‐differentiability in the common sense of state variables in multimodal critical states, are overcome by means of the Lagrange multiplier method. Further constraints impose that only the first critical equilibrium states (local maxima or bifurcation points) on the initial equilibrium path of the actual designs are taken into account. Copyright © 2002 John Wiley & Sons, Ltd.

Optimum design of truss structures undergoing large deflections subject to a system stability constraint

A structural optimization algorithm is developed for shallow trusses undergoing large deflections subject to a system stability constraint. The method combines the non-linear buckling analysis, through displacement control technique, with the optimality criteria approach. Four examples illustrate the procedure and allow the results obtained to be compared with those in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass for these types of structures so a design based on the linear buckling analysis can result in catastrophic failure. In one of the design examples the stresses in the elements, in the optimum design, exceed the allowable stresses, pointing out the need for a design that accounts for both non-linear buckling and stress constraints. Copyright © 2000 John Wiley & Sons, Ltd.

Local stability of trusses in the context of topology optimization Part I: Exact modelling

The paper considers the problem of optimal truss topology design with respect to stress and local stability (i.e. buckling) constraints. In a context of topology optimization, the exact. management of buckling constraints is highly complex: member forces must satisfy functions which discontinuously depend on the design variables.