Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

Topology optimization with design-dependent pressure loading

In this paper, the layout of structures under design-dependent pressure loading is optimized using a topology optimization approach. In contrast to topology optimization problems with conventional static external loading, the position and direction of pressure loading are changing with topology of structure during optimization iterations. In order to model the changing structural surface boundaries under design-dependent pressure loading, a pseudo equal-potential function is introduced. Design sensitivity analysis is derived from the adjoint method. Three examples solved by the proposed method are presented.     

Topology optimization of piezoelectric smart structures for minimum energy consumption under active control

This paper investigates topology optimization of the electrode coverage over piezoelectric patches attached to a thin-shell structure to reduce the energy consumption of active vibration control under harmonic excitations. The constant gain velocity feedback control method is employed, and the structural frequency response under control is analyzed with the finite element method. In the mathematical formulation of the proposed topology optimization model, the total energy consumption of the control system is taken as the objective function, and a constraint of the maximum allowable dynamic compliance is considered. The pseudo-densities indicating the distribution of surface electrode coverage over the piezoelectric layers are chosen as the design variables, and a penalized model is employed to relate the active damping effect and these design variables. The sensitivity analysis scheme of the control energy consumption with respect to the design variables is derived with the adjoint-variable method. Numerical examples demonstrate that the proposed optimization model is able to generate optimal topologies of electrode coverage over the piezoelectric layers, which can effectively reduce the energy consumption of the control system. Also, numerical comparisons with a minimum-volume optimization model show the advantage of the proposed method with respect to energy consumption. The proposed method may provide useful guidance to the layout optimization of piezoelectric smart structures where the energy supply is limited, such as miniature vibration control systems.     

Combined optimization of bi-material structural layout and voltage distribution for in-plane piezoelectric actuation

This paper investigates the combined optimization of bi-material structural layout and actuation voltage distribution of structures with embedded in-plane piezoelectric actuators. The maximization of the nodal displacement at a selected output port is considered as the design objective. A two-phase material model with power-law penalization is employed in the topology optimization of the actuator elements and the coupled surrounding structure. In order to incorporate the actuation voltage directly into the design for achieving the best overall actuation performance, element-wise voltage design variables are also included in the optimization. For the purpose of easy implementation of the electric system, the allowable voltage levels at an individual element are confined to three discrete values, namely zero and two prescribed values with opposite signs. To this end, a special interpolation scheme between the tri-level voltage values and the design variables is used in the optimization model. Based on the design sensitivity analysis of the objective function, the combined optimization problem is solved with the MMA algorithm. Numerical examples are presented to demonstrate the applicability of the proposed optimization model and numerical techniques. The optimal solutions also confirmed that larger output displacement can be achieved by introducing voltage design variables into the design problem.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

Simultaneous design of components layout and supporting structures using coupled shape and topology optimization technique

The purpose of this paper was to study the layout design of the components and their supporting structures in a finite packing space. A coupled shape and topology optimization (CSTO) technique is proposed. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a finite-circle method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. As the FE mesh discretizing the packing space, i.e., design domain, has to be updated itertively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain will be introduced in this paper instead of using traditional pseudo-density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure, and design sensitivities are calculated w.r.t both geometric variables and density variables. In this paper, several design problems maximizing structural stiffness are considered subject to the material volume constraint. Reasonable designs of components layout and supporting structures are obtained numerically.     

Topology optimization of flexible micro-fluidic devices

A multi-objective topology optimization formulation for the design of dynamically tunable fluidic devices is presented. The flow is manipulated via external and internal mechanical actuation, leading to elastic deformations of flow channels. The design objectives characterize the performance in the undeformed and deformed configurations. The layout of fluid channels is determined by material topology optimization. In addition, the thickness distribution, the distribution of active material for internal actuation, and the support conditions are optimized. The coupled fluid-structure response is predicted by a non-linear finite element model and a hydrodynamic lattice Boltzmann method. Focusing on applications with low flow velocities and pressures, structural deformations due to fluid-forces are neglected. A mapping scheme is presented that couples the material distributions in the structural and fluid mesh. The governing and the adjoint equations of the resulting fluid-structure interaction problem are derived. The proposed method is illustrated with the design of tunable manifolds.     

Automotive applications of topology optimization

Topology optimization is used for obtaining the best layout of vehicle structural components to achieve predetermined performance goals. An in-house topology optimization software, TOP, has been developed to analyse important automotive components. The topology design problem is formulated as a general optimization problem and is solved by the mathematical programming method. The MSC/NASTRAN finite element code is employed for response analyses. The use of MSC/NASTRAN is significant, because it not only allows engineers to use a wellaccepted and widely-used finite element code with no size limit on the model, but also permits developers to concentrate on the rest of the topology optimization program. Three automotive examples including a simplified truck frame, a deck lid, and a space frame structure are presented.     

Robust topology optimization for dynamic compliance minimization under uncertain harmonic excitations with inhomogeneous eigenvalue analysis

Variability of load magnitude/direction is a most significant source of uncertainties in practical engineering. This paper investigates robust topology optimization of structures subjected to uncertain dynamic excitations. The unknown-but-bounded dynamic loads/accelerations are described with the non-probabilistic ellipsoid convex model. The aim of the optimization problem is to minimize the absolute dynamic compliance for the worst-case loading condition. For this purpose, a generalized compliance matrix is defined to construct the objective function. To find the optimal structural layout under uncertain dynamic excitations, we first formulate the robust topology optimization problem into a nested double-loop one. Here, the inner-loop aims to seek the worst-case combination of the excitations (which depends on the current design, and is usually to be found by a global optimization algorithm), and the outer-loop optimizes the structural topology under the found worst-case excitation. To tackle the inherent difficulties associated with such an originally nested formulation, we convert the inner-loop into an inhomogeneous eigenvalue problem using the optimality condition. Thus the double-loop problem is reformulated into an equivalent single-loop one. This formulation ensures that the strict-sense worst-case combination of the uncertain excitations for each intermediate design be located without resorting to a time-consuming global search algorithm. The sensitivity analysis of the worst-case objective function value is derived with the adjoint variable method, and then the optimization problem is solved by a gradient-based mathematical programming method. Numerical examples are presented to illustrate the effectiveness and efficiency of the proposed framework.     

Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.     

Multidiscipline topology optimization

Topology optimization is used for determining the best layout of structural components to achieve predetermined performance goals. The density method which uses material density of each finite element as the design variable, is employed. Unlike the most common approach which uses the optimality criteria methods, the topology design problem is formulated as a general optimization problem and is solved by the mathematical programming method. One of the major advantages of this approach is its generality; thus it can solve various problems, e.g. multi-objective and multi-constraint problems. In this study, the structural weight is chosen as the objective function and structural responses such as the compliances, displacements and the natural frequencies, are treated as the constraints. The MSC/NASTRAN finite element code is employed for response analyses. One example with four different optimization formulations was used to demonstrate this approach.     

Adaptive topology optimization

Topology optimization of continuum structures is often reduced to a material distribution problem. Up to now this optimization problem has been solved following a rigid scheme. A design space is parametrized by design patches, which are fixed during the optimization process and are identical to the finite element discretization. The structural layout is determined, whether or not there is material in the design patches. Since many design patches are necessary to describe approximately the structural layout, this procedure leads to a large number of optimization variables. Furthermore, due to a lack of clearness and smoothness, the results obtained can often only be used as a conceptual design idea.To overcome these shortcomings adaptive techniques, which decrease the number of optimization variables and generate smooth results, are introduced. First, the use of pure mesh refinement in topology optimization is discussed. Since this technique still leads to unsatisfactory results, a new method is proposed that adapts the effective design space of each design cycle to the present material distribution. Since the effective design space is approximated by cubic or Bézier splines, this procedure does not only decrease the number of design variables and lead to smooth results, but can be directly joined to conventional shape optimization. With examples for maximum stiffness problems of elastic structures the quality of the proposed techniques is demonstrated.     

An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems

The conventional coupled sensitivity analysis method for concurrent topology optimization problems is computationally expensive for microscale design variables. This study thus proposes an efficient decoupled sensitivity analysis method for concurrent topology optimization based on the chain differentiation rule. Two numerical studies are performed to demonstrate the effectiveness of the decoupled sensitivity analysis method for concurrent topology optimization problems with single or multiple porous materials. It can be concluded from the results that the decoupled method is computationally much more efficient than the coupled method, while they are mathematically equivalent. The outstanding merits of the decoupled method are two-fold: (1) computational efficiency of sensitivity analysis with respect to the microscale design variables; and (2) applicability to concurrent topology optimization problems with single or multiple porous materials as well as with composite microstructure and multi-phase materials.     

Maximization of structural natural frequency with optimal support layout

The optimal layout of supports is one of the key factors that dominates static and dynamic performances of the structure. In this work, supports are considered as elastic springs. The purpose is to carry out layout optimization of supports by means of topology optimization method. The technique of pseudo-density variables that transforms a discrete-variable problem into a continuous one is used in order that the problem is easily formulated and solved numerically. In this formulation, a power law of the so-called solid isotropic material with penalty model is employed to approximate the relation between the element stiffness matrix and density variable. Such a relation makes it easy to establish the computing scheme and sensitivity analysis of natural frequency. Support layout design that corresponds to optimization of boundary conditions is studied to maximize the natural frequency of structures. Numerical results show that reasonable distributions of supports can be generated effectively.     

Adaptive topology optimization of elastoplastic structures

Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.     

Adaptive gradient-enhanced kriging model for variable-stiffness composite panels using Isogeometric analysis

A vibration isolation system is designed using novel hybrid optimization techniques, where locations of machines, locations of isolators and layout of supporting structure are all taken as design variables. Instead of conventional parametric optimization model, the 0-1 programming model is established to optimize the locations of machines and isolators so that the time-consuming remeshing procedure and the complicated sensitivity analysis with respect to position parameters can be circumvented. The 0-1 sequence for position design variables is treated as binary bits so as to reduce the actual number of design variables to a great extent. This way the 0-1 programming can be solved in a quite efficient manner using a special version of genetic algorithm(GA) that has been published by the authors. The layout of supporting structure is optimized using SIMP based topology optimization method, where the fictitious elemental densities are taken as design variables ranging from 0 to 1. Influence of different design variables is firstly investigated by numerical examples. Then a hybrid multilevel optimization method is proposed and implemented to simultaneously take all design variables into account.     

A mathematical model of higher level structural optimization problems and their solution

In this paper, a new mathematical model suitable for higher level structural optimization problems, such as optimization of structural topology, layout and type is presented. In this mathematical model, the relation between two structures with different layouts is established by introducing the nonbasic variables. Using the Kuhn-Tucker condition for optimality, a criterion for determining a better layout of a structure is developed. This provides a measure for selecting the optimal layout of a structure. The method introduces a new way for higher level structural optimization design. Several numerical examples are given to illustrate the effectiveness of this method.     

Topology optimization of three-dimensional structures using optimum microstructures

In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium.The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem.Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.