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Topology optimization of transient dynamics is of great importance in engineering. However, the computational cost required by the repeated dynamic response and design sensitivity analysis during design iterations is often extremely high for large-scale problems, which significantly limits its practical application. To address this issue, we propose a new method based on successive iteration of analysis and design in conjunction with a quasi-static response-enhanced mode displacement method. The conventional mode displacement method requires an inner-loop iteration in addition to the outer-loop design updating, for finding the eigenmodes of the undamped system. In order to avoid the high computational burden caused by the nested double-loop solution, we employ the concept of successive iteration of analysis and design, which integrates approximate eigenvalue analysis and design variable updating into a single iteration loop. By this means, the eigenmodes are sequentially improved through a one-step inverse iteration in each design iteration, and simultaneous convergence of the eigenmodes and the design is sought during the optimization process. These approximate eigenmodes and the quasi-static response are used as the model reduction basis for each intermediate design, and the direct time integration of the reduced-dimensional system is then performed to calculate the structural responses of interest and their derivatives with respect to the design variables. Several numerical examples are presented to demonstrate the efficiency of the proposed method. It is shown that the method can be used to solve transient dynamic topology optimization problems up to 15 million degrees of freedom on a desktop computer at affordable time cost. [ABSTRACT FROM AUTHOR]

Topology optimization has become a useful tool for designing Phononic Crystals (PnCs) to achieve desired elastic wave propagation properties. However, topological design of three-dimensional (3D) PnCs remains a significant challenge due to extremely high computational cost required for the repeated solution of the nested double-loop problem of design optimization and eigenvalue analysis to determine the band structure during the optimization iteration process. This paper presents a more efficient topology optimization method based on the concept of successive iteration of analysis and design. The method employs a sequential approximation of the eigenpairs through an inverse iteration-like procedure to avoid solving the expensive eigenvalue problems in each design iteration. To further improve the efficiency of the design iteration, the eigenmodes of the intermediate PnCs designs are first computed on a relatively coarse finite element mesh. Then they are mapped onto the fine mesh and used as the initial trial eigenmodes in the eigenpair analysis and design sensitivity analysis. The high efficiency of the proposed method, as compared with conventional nested double-loop optimization approaches, is demonstrated by 3D numerical examples with over one million degrees of freedom. It is shown that this approach can achieve simultaneous convergence of the eigenvectors along with the design evolution of the unit cell, therefore substantially reduces the computational burden for high-resolution topological design of 3D PnCs. [ABSTRACT FROM AUTHOR]

In this paper, the layout optimization problem of continuum structures embedded with movable components and holes simultaneously is solved for the first time. We propose a new methodology for the embedding problem under SIMP-based computational framework, where the positions and orientations of embedded components and holes and the topology of connecting structure are optimized concurrently to maximize the overall stiffness. To this end, the parameterized topology description function, combining the Kreisselmeier-Steinhauser (KS) function, is used to construct the geometric shapes of embedded components and holes. The material density defining the topology of connecting structure and the geometric parameters used to describe the location and orientation of embedded components and holes are considered as design variables of the optimization problem. To unite these two seemingly different representations into a single computational framework, we first project the embedded components and holes into two density fields on a fixed grid using a smoothed Heaviside function, then introduce a new SIMP-motivated material interpolation scheme invoked at the finite element level for material parameterization. The material parameterization scheme supports full analytical sensitivities, which can greatly improve the accuracy and efficiency of sensitivity analysis, and make it suitable for use with efficient gradient-based algorithms. Finally, the effectiveness of the proposed method is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the sensitivity analysis of structural acoustic performance in presence of non-proportional damping and optimal layout design of the damping layer of vibrating shell structures under harmonic excitations. The structural system with a partially-covered damping layer has a non-proportional global damping matrix. Therefore, the method of complex mode superposition in the state space is employed in the dynamic response analysis. The sound pressure is calculated with the structural response solution by using the boundary element method. In this context, an adjoint variable scheme for the design sensitivity analysis of sound pressure is developed. In the optimal design problem, the design objective is to minimize the structural vibration-induced sound pressure at a specified point in the acoustic medium by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is employed, and the relative densities of the damping material are considered as design variables. Numerical examples are given to illustrate the validity and efficiency of this approach. The influences of the excitation frequency, the damping coefficients and the locations of the reference point on the optimal topologies are also discussed. [Copyright &y& Elsevier]

The sensitivity analysis of rigid viscoplastic deformation processes with application to metal preform design optimization is investigated. For viscoplastic constitutive models, the deformation process is path-dependent in nature and thus the sensitivity analysis of the deformation history is formulated in an incremental procedure. To this end, an algorithm is derived on the basis of the time integration scheme used in the primary finite element analysis, where the contact conditions are treated with the penalty method. The discretized equilibrium equations, as well as the time integration equations, are directly differentiated with respect to the design variables. The discrete form of the sensitivity equations is then solved with procedures similar to those used in the direct analysis, where the secant matrix decomposed in the direct analysis can also be utilized at each time instant. Thus the sensitivity of the deformation history is evaluated in a step-wise procedure. The present algorithm can be employed for the optimization of metal forming processes. The accuracy of the proposed sensitivity analysis as well as its applicability are demonstrated by numerical examples with reference to preform design optimization problems, where the aggregate function method is employed for converting the non-smooth Min–max type objective function into a numerically tractable one. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an efficient topology optimization strategy for seeking the optimal layout of continuum structures exhibiting asymmetrical strength behaviors in compression and tension. Based on the Drucker–Prager yield criterion and the power-law interpolation scheme for the material property, the optimization problem is formulated as to minimize the material volume under local stress constraints. The ε-relaxation of stress constraints is adopted to circumvent the stress singularity problem. For improving the computational efficiency, a grouped aggregation approach based on the Kreisselmeier–Steinhauser function is employed to reduce the number of constraints without much sacrificing the approximation accuracy of the stress constraints. In conjunction with the adjoint-variable sensitivity analysis, the minimization problem is solved by a gradient-based optimization algorithm. Numerical examples demonstrate the validity of the present optimization model as well as the efficiency of the proposed numerical techniques. Moreover, it is also revealed that the optimal design of a structure with pressure-dependent material may exhibit a considerable different topology from the one obtained with pressure-independent material model. [Copyright &y& Elsevier]

This article presents a study on topology optimization of planar piezoelectric actuators assembled with repetitive component patterns. Repetitive configuration has the advantage of ease of engineering implementation, especially for relatively slender structures. For realizing this concept in the design of piezoelectric actuators, topology optimization techniques are employed for seeking the optimal layout within the design domain of the structural components. The design objective is to maximize the work delivered by the displacement output port, while constraints are imposed on the actuation energy consumption and the material volume. Both the distributions of the actuation voltage and the topologies of the host layer and the piezoelectric layers are optimized. Power-law penalization functions are used to suppress intermediate values of material densities and actuation voltage. Numerical techniques for a sensitivity analysis of structural response are presented, and the proposed optimization problem is solved with a gradient-based mathematical programming approach. Two numerical examples are given to demonstrate the applicability of the proposed approach. Compared with the method directly treating the whole design domain, this approach is shown to have a better convergence behavior and is able to provide final topologies that are better acceptable from engineering point of view. [ABSTRACT FROM PUBLISHER]

Abstract: This paper investigates the formulation and numerical solution of reliability-based optimization of structures exhibiting grouped uncertain-but-bounded variations. Based on the multi-ellipsoid convex model description for grouped uncertain-but-bounded parameters, the mathematical definition of a non-probabilistic reliability index is presented for quantified measure of the safety margin. The optimal design is then formulated as a nested optimization problem. A method based on concerned performance is proposed for regularization of the reliability index constraints. The expensive computation of the non-probabilistic reliability index and its derivative is thus avoided. Numerical examples are given to illustrate the validity and efficiency of the present method. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract  Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization. [ABSTRACT FROM AUTHOR]

Abstract: The robust design of non-linear structures with path-dependent response is stated as a two-criteria optimization problem and is solved by the method of mathematical programming. To this end, the perturbation technique is applied in conjunction with the incremental loading procedure for the response moment analysis of path-dependent non-linear structural systems with random parameters. Furthermore, the sensitivities of mean and variance of the structural performance function are evaluated using direct differentiation in the framework of perturbation based stochastic finite element analysis. By introducing a weighting factor in the compound objective––resp. desirability function, and feasibility indices in the constraints, the mathematical model of structural robust design problem is formulated and is solved with a gradient-based algorithm. Numerical examples demonstrate the applicability of the presented method. [Copyright &y& Elsevier]

The robust design of structures with stochastic parameters is studied using optimization techniques. The first two statistical moments of the stochastic parameters including design variables are considered in conjunction with the second-order perturbation method for the approximation of mean value and variance of the structural response. In this framework, the sensitivities of the mean values and variances of the structural performance function with respect to the design variables are calculated for use in the optimization task. The robust design of structures is formulated as a multi-criteria optimization problem, in which both the expected value and the standard deviation of the objective function are to be minimized. The robustness of the feasibility is also taken into account by involving the variability of the structural response in the constraints. The two-criteria optimization problem is converted into a scalar one and is then solved by a gradient based optimization algorithm. To demonstrate the applicability of the presented method, numerical examples are given, involving static and dynamic response. [Copyright &y& Elsevier]

This paper investigates the simultaneous optimal distribution of structural material and trilevel actuation voltage for static shape control applications. In this optimal design problem, the shape error between the actuated and the desired shapes is chosen as the objective function. The energy and the material volume are taken as constraints in the optimization problem formulation. The discrete-valued optimization problem is relaxed using element-wise continuous design variables representing the relative material density and the actuation voltage level. Artificial interpolation models which relate the mechanical/piezoelectrical properties of the material and the actuation voltage to the design variables are employed. Therein, power-law penalization functions are used to suppress intermediate values of both the material densities and the control voltage. The sensitivity analysis procedure is discussed, and the design variables are optimized by using the method of moving asymptotes (MMA). Finally, numerical examples are presented to demonstrate the applicability and effectiveness of the proposed method. It is shown that the proposed method is able to yield distinct material distribution and to suppress intermediate actuation voltage values as required. [ABSTRACT FROM AUTHOR]

This paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density-based topology optimization of elastic structures undergoing large displacements. By adding a special hyperelastic material to the design domain, excessive distortion and numerical instability occurred in the low-density or intermediate-density elements are thus effectively alleviated during the optimization process. The properties of the additional hyperelastic material are established based on a new interpolation scheme, which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure. In conjunction with the adjoint variable scheme for sensitivity analysis, the topology optimization problem is solved by a gradient-based mathematical programming algorithm. Numerical examples are given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a new Multi-Material Level Set (MM-LS) topology description model for topology and shape optimization of structures involving multiple materials. Each phase is represented by a combined formulation of different level set functions. With a total number of M level set functions, this approach provides a representation of M materials and one void phase (totally M + 1 phases). The advantages of the proposed method include: (1) it can guarantee that each point contains exactly one phase, without overlaps between each two phases and redundant regions within the design domain; (2) it possesses an explicit mathematical expression, which greatly facilitates the design sensitivity analysis; and (3) it retains the merits of the level set method, including smooth boundary and distinct interface. A parametric level set method is applied to evolve the topology and shape of multi-material structures, with a high computational efficiency. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

• An effective method is developed for minimum stress design with movable holes. • The moving holes are projected onto a fixed grid to avoid remeshing the grids. • New stiffness and stress penalizations are proposed for the considered problem. • The proposed method performs well in minimum stress design with movable holes. Currently, most works on layout optimization problem of continuum structure embedded with movable holes are all carried out to maximize the stiffness of the overall system. In this work, the embedding problem is solved for minimum stress design for the first time. To this end, we propose an effective hybrid methodology under SIMP-based computational framework. The material density characterizing the topology of the load transfer path and the geometric parameters defining the rigid body motion (translation and rotational) of the embedded holes are considered as design variables and are optimized simultaneously to minimize the aggregated maximum stress. To unite these two seemingly different representations into an optimization model, we mapped the embedded holes into a density field on a fixed grid using a Sigmoid activation function. Then, a new SIMP-like material interpolation scheme that considers embedding holes is introduced for stiffness penalization and stress penalization. The optimization model for solving the minimum stress problem embedded with movable holes and its sensitivity analysis are detailed. Finally, two typical examples are performed to illustrate the effectiveness of the proposed methodology. [ABSTRACT FROM AUTHOR]

In the semi-active vibration control of structural systems, best control performance and low energy consumption both play important roles. This paper presents a new topology optimization method for the multi-electrode layout design of sandwich electrorheological (ER) plates considering limited energy consumption under different vibration conditions. In the proposed topology optimization model, the dynamic compliance under specified harmonic excitation is taken as the objective function, and a constraint of the maximum allowable energy consumption is considered. The frequency response analysis is implemented with finite element discretization, and the sensitivity analysis scheme of the control energy consumption for the design variables is derived with the adjoint-variable method. Numerical examples showed that optimal layouts of multi-electrodes could be obtained with different energy consumption requirements and loading conditions by using the proposed method. The excellent performance of vibration control could be achieved by only varying the voltage distribution of ERs. • A new topology optimization method for multi-electrode layout design of ER plates is proposed. • Energy consumption constraint is considered in the vibration optimization model. • Best vibration control performance can achieved by only switching the optimal electrode layouts. • Frequency response sensitivity analysis for ER plate is derived with adjoint variable method. [ABSTRACT FROM AUTHOR]