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Designing multichannel phononic crystal (PnC) resonant cavities working at certain frequencies is highly nontrivial because the defect band location within the band gap must be precisely located. This paper discloses a new methodology of the topological design for the precisely‐controlled PnC resonant cavity. A two‐stage topological design strategy is proposed to determine the frequency ranges of the forbidden and defect bands successively. Complicated topologies of the supercell are represented only with dozens of design variables based on material‐field series expansion model and can be optimized with a gradient‐free optimization algorithm. Different PnC resonant cavity configurations that are difficult to obtain through traditional design methods are realized through the proposed strategy. A multichannel energy harvester is thus designed with the optimized supercell configurations. The device demonstrates a high localization effect (>300%), high frequency resolution of the incident wave (≈0.2 kHz), and a 2 mW power output. [ABSTRACT FROM AUTHOR]

Abstract This paper studies concurrent two-scale design optimization of composite structures filled with multiple microstructural unit cells. The task of the design problem is to simultaneously optimize microstructural configurations of the unit cells and their spatial distribution in the macroscale. To this end, a new topology optimization framework based on combined topology representation of the density model and the level set model is proposed. The homogenization method is used to link the material microstructural design and the macroscale design by evaluating the effective properties of the microstructures. In the microscale, topology optimization of multiple microstructural unit cells is performed with the density-based method. In the macroscale design, the distribution of multiple microstructural unit cells is optimized by the velocity field level set method, which inherits advantages of the implicit geometrical representation of the conventional level set model (relatively clear and smooth material boundaries/interfaces, more natural description of topological evolution). Moreover, the velocity field level set method maps the variational boundary shape optimization problem into a finite-dimensional design space, thus making it relatively easy and efficient to employ general mathematical programming algorithms to handle the multiple constraints and two types of design variables in the concurrent two-scale design problem. Numerical examples show that the present concurrent two-scale design method can generate meaningful designs of hierarchical cellular structures with well-defined boundaries and material interfaces. Highlights • A concurrent two-scale design of multiple unit cells and structure is presented. • VFLS and SIMP method are respectively used in macroscale and microscale designs. • Level set model gives a rigorous interpretation of microstructure distribution. • VFLS method can efficiently handle multiple constraints in multiscale design. • Using multiple unit cells can obtain higher structural stiffness than uniform ones. [ABSTRACT FROM AUTHOR]

Abstract Material interface-related uncertainties induced by inter-diffusion or reactions between two different materials may deteriorate the actual performance of a structural design achieved by topology optimization. Thus a rational methodology is needed to address this issue in the design of hybrid-material engineering products implemented by some novel fabrication techniques such as additive manufacturing. This paper presents a robust shape and topology optimization method accounting for uncertain graded interface properties of multi-material structures. A level set function is used to track the evolving material interfaces during the optimization process, and the material interface uncertainties is modeled by introducing an intermediate zone with graded properties represented by a random field. On the basis of discretizing the input random field by means of the Expansion Optimal Linear Estimation (EOLE) method, the uncertain propagation analysis is implemented with the Polynomial Chaos expansion (PCE) to predict the stochastic response. Then the robust shape and topology optimization problem is stated as a multi-criteria optimization problem, in which the expected value and the standard deviation of the performance function of interest are to be minimized under a given material volume constraint. The shape derivative of the stochastic response is derived in the context of Eulerian description, and then used to advance the evolution of the level set function through the Hamilton-Jacobi equation. In the numerical examples, the proposed robust design method is exemplified by the mean compliance minimization problems. [ABSTRACT FROM AUTHOR]

It is often highly desirable to simultaneously optimize the layout of embedded functional components and the topology of the host structure supporting these components to achieve the best overall performance while still ensuring the structural integrity. We propose a topology optimization framework to account for connecting interface behaviors between the components and the host structure. Here we treat the connecting interfaces with the cohesive zone model to reflect the adhesively bonded interface behaviors. A conforming mesh in conjunction with interface elements is employed to discretize the evolving structure while accounting for the strong discontinuity of displacement field across the material interfaces. To give a clear representation of structural boundaries and the connecting interfaces, we also suggest a multi-material interpolation model in the level set framework, which can conveniently define the connecting interface locations and describe multi-material distribution without redundant phase in the design domain. The objective function is defined as the sum of the strain energy and the work done by the traction on the connecting interface, and the evolution velocities of the level set and the embedded components are treated as design variables. These design variables are updated with the MMA optimizer on the basis of adjoint-variable sensitivity analysis. This optimization formulation allows multiple constraints and mixed design variables ( i.e level set design variables and components’ design variables) to be easily handled in level set based optimization. The design is advanced by the Hamilton–Jacobi equation with the velocity design variables as input. Numerical examples demonstrate the validity and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Summary: This paper develops a new reliability‐based topology optimization framework considering spatially varying geometric uncertainties. Geometric imperfections arising from manufacturing errors are modeled with a random threshold model. The projection threshold is represented by a memoryless transformation of a Gaussian random field, which is then discretized by means of the expansion optimal linear estimation. The structural response and their sensitivities are evaluated with the polynomial chaos expansion, and the accuracy of the proposed method is verified by Monte Carlo simulations. The performance measure approach is adopted to tackle the reliability constraints in the reliability‐based topology optimization problem. The optimized designs obtained with the present method are compared with the deterministic solutions and the reliability‐based design considering random variables. Numerical examples demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Manufacturing-oriented topology optimization has been extensively studied the past two decades, in particular for the conventional manufacturing methods, for example, machining and injection molding or casting. Both design and manufacturing engineers have benefited from these efforts because of the close-to-optimal and friendly-to-manufacture design solutions. Recently, additive manufacturing (AM) has received significant attention from both academia and industry. AM is characterized by producing geometrically complex components layer-by-layer, and greatly reduces the geometric complexity restrictions imposed on topology optimization by conventional manufacturing. In other words, AM can make near-full use of the freeform structural evolution of topology optimization. Even so, new rules and restrictions emerge due to the diverse and intricate AM processes, which should be carefully addressed when developing the AM-specific topology optimization algorithms. Therefore, the motivation of this perspective paper is to summarize the state-of-art topology optimization methods for a variety of AM topics. At the same time, this paper also expresses the authors’ perspectives on the challenges and opportunities in these topics. The hope is to inspire both researchers and engineers to meet these challenges with innovative solutions. [ABSTRACT FROM AUTHOR]

This article investigates topology optimization for piezoelectric thin-shell structures under the linear quadratic regulator optimal control. In the optimization model, the structural dynamic compliance is taken as the measure of control performance, and the relative densities describing the distribution of the piezoelectric material are considered as design variables. An artificial material model with penalization on both mechanical and piezoelectric properties is employed. For the purpose of improving computational efficiency of the sensitivity and response analysis, modal superposition method is adopted. The derivative of the Riccati equation governing the linear quadratic regulator control with respect to the design variables is shown to be a Lyapunov equation. In conjunction with the adjoint variable method, the design sensitivities of the dynamic compliance are obtained using the solution of the Lyapunov equation. Numerical examples demonstrate the validity of the proposed method and show the significance of layout design of piezoelectric sensors/actuators. The influences of some key factors on the optimization solutions are discussed. It is shown that the optimized layout of the piezoelectric patches may be significantly influenced by the excitation frequency, but only slightly affected by the choice of the weighting matrix in the linear quadratic regulator control. This work aims to provide an efficient gradient-based mathematical programming method for guiding the layout design of actuators and sensors in smart structures under optimal vibration control. However, the considered model is a purely mathematical one without consideration of engineering realization, thus the optimization result may only serve as an upper bound for practically realizable control performance. [ABSTRACT FROM AUTHOR]

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications. [ABSTRACT FROM AUTHOR]

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 ɛ-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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

As a set theory-based convex model, the ellipsoidal model provides an attractive framework for treating uncertain-but-bounded variations in the structural reliability analysis and design optimization. However, improper modeling of the uncertainties may give rise to misleading non-probabilistic reliability analysis, thus result in either unsafe or over-conservative designs. This paper presents a systematic study on the mathematical formulation for constructing the minimum-volume ellipsoidal convex model using a given set of sample data, and shows its application in existing methods of non-probabilistic reliability analysis and design optimization of structures with bounded uncertainties. In this method, the uncertain parameters are first divided into groups according to their sources. For each individual group of uncertainties, the minimum-volume ellipsoid problem is reformulated into a semi-definite programming (SDP) problem and thus can be efficiently solved to its global optimum. Further, a linear transformation based on the eigenvalue analysis is employed to map the ellipsoidal model into a standard uncertainty space. This uncertainty modeling technique enables a compact and differentiable bound description of the parameter variations. Moreover, it has another useful property, the affine invariance, which is shown to be necessary for meaningful definition of a non-probabilistic reliability index. The effectiveness and efficiency of the present techniques for convex model construction and the corresponding reliability analysis are demonstrated with numerical examples of structural topology optimization problems with bounded variations arising from different sources. [ABSTRACT FROM AUTHOR]

It is impractical to implement arbitrary-shaped piezoelectric patches from the view point of manufacturability of fragile piezoelectric ceramics, thus using designable electrode layers to deliver desired actuation forces provides a more realistic option in engineering applications. This study develops a topological design method of surface electrode distribution over piezoelectric sensors/actuators attached to a thin-walled shell structure for reducing the sound radiation in an unbounded acoustic domain. In the optimization model, the sound pressure norm at specific reference points under excitations at a certain excitation frequency or in a given frequency range is taken as the objective function. The pseudo densities for indicating absence and presence of surface electrodes at each element are taken as the design variables, and a penalized relationship between the densities and the active damping effect is employed. The vibrating structure is discretized with finite element model for the frequency response analysis and the sound radiation analysis in the unbounded acoustic domain is treated by boundary element method. The applied voltage on each actuator is determined by the constant gain velocity feedback (CGVF) control law. The technique of the complex mode superposition in the state space, in conjunction with a model reduction transformation, is adopted in the response analysis of the system characterized by a non-proportional active damping property. In this context, the adjoint-variable sensitivity analysis scheme is derived. The effectiveness and efficiency of the proposed method are demonstrated by numerical examples, and several key factors on the optimal designs are also discussed. [ABSTRACT FROM AUTHOR]

This paper investigates topology optimization of the piezoelectric actuator/sensor coverage attached to a thin-shell structure to improve the active control performance for reducing the dynamic response under transient excitations. The constant gain velocity feedback (CGVF) control algorithm is considered and the structural dynamic response under the corresponding active damping effect is evaluated with a direct time integration method. In the mathematical formulation of the considered topology optimization model, the time integral of the displacement response over a specified time interval of interest is taken as the objective function. The pseudo-densities describing the piezoelectric material distribution are taken as the design variables, and a penalization model on the structural stiffness and piezoelectric effect is employed. The adjoint-variable sensitivity analysis scheme for a general integral function within a given time interval is derived, which facilitates a gradient-based mathematical programming solution of the optimization problem. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of piezoelectric layers. Also, the influences of the control gain and the integration time intervals in the objective function on the optimal solutions are discussed. The proposed method can be used for providing useful guidance to the layout design of the actuator/sensor layers attached to a thin-shell structure subject to dynamic excitations, in particular impact forces. It is also confirmed that the achieved vibration reduction is mainly due to improvement of the active control performance rather than changes of the structural dynamic stiffness/mass property. [ABSTRACT FROM AUTHOR]

Highlights: [•] A new adaptive refinement strategy for topology optimization is proposed. [•] Independent error controls applied to separated displacement and density fields. [•] Analysis mesh is refined to improve computational accuracy within local regions. [•] Density field is refined to improve boundary description quality. [•] High-accuracy and high-quality solutions at less cost than fixed mesh approach. [ABSTRACT FROM AUTHOR]

This article investigates topology optimization of the piezoelectric actuator and sensor layers in a plate for achieving the best vibration control performance. Therein, the actuator patches and sensor patches are symmetrically attached to the host layer, and the classical negative velocity feedback control strategy is adopted for reducing the vibration level of the structure. In the optimization model, the dynamic compliance under a specific excitation frequency or the aggregated dynamic compliance in a given frequency range is taken as the objective function. The relative densities of the elements in the actuator layer and the sensor layer are considered as topological design variables. The optimization problem is then formulated by using an artificial material model with penalization for both mechanical and piezoelectric properties. It is pointed out that the global-level damping property, consisting of the structural damping and the active damping effects, is a nonproportional one. For alleviating the computational burden involved in the frequency response analysis, the dynamic equations are solved with the complex mode superposition in the state space after a model reduction transformation. In this context, the sensitivity analysis scheme is also derived. The effectiveness and efficiency of the proposed method are demonstrated by 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]

Abstract: This paper presents an adaptive density point refinement approach for continuum topology optimization on the basis of an analysis-mesh separated material density field description based on nodal design variables. The Shepard interpolants are used to construct a strictly range-restricted density field over the design domain with the density design variables defined on a density point grid. Since the density points are defined independent of the finite element mesh, it is easy to refine the density point grid without remeshing the finite element model. A refinement criterion is given to identify the gray transitional regions to be adaptively refined in the subsequent optimization iterations. With such a refinement scheme, the topology optimization can start from a relatively coarse density point grid but still yields a desired higher resolution of the structural boundaries in the final design. Because refinements are only performed when and where necessary, this method is able to improve the boundary description quality of the optimal result with much less design variables as compared with the case of global refinement, and therefore can greatly reduce the computational burden involved in the sensitivity analysis and optimization process. Moreover, the percentage of transitional regions in the final solutions can also be reduced. Compared with using a uniformly globally-dense density point arrangement, this approach can achieve similar optimal designs but with much less computational cost. Numerical examples are given to demonstrate the effectiveness and efficiency of the present approach. [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]

Abstract: This paper presents a non-local density interpolation strategy for topology optimization based on nodal design variables. In this method, design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes. By using the Shepard family of interpolants, the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point. The employed interpolation scheme has an explicit form and satisfies non-negative and range-restricted properties required by a physically significant density interpolation. Since the discretizations of the density field and the displacement field are implemented on two independent sets of points, the method is well suited for a topology optimization problem with a design domain containing higher-order elements or non-quadrilateral elements. Moreover, it has the ability to yield mesh-independent solutions if the radius of the influence domain is reasonably specified. Numerical examples demonstrate the validity of the proposed formulation and numerical techniques. It is also confirmed that the method can successfully avoid checkerboard patterns as well as “islanding” phenomenon. [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: This paper describes a non-probabilistic reliability-based topology optimization method for the design of continuum structures undergoing large deformations. The variation of the structural system is treated with the multi-ellipsoid convex model, which is a realistic description of the parameters being inherently uncertain-but-bounded or lacking sufficient probabilistic data. The formulation of the optimal design is established as a volume minimization problem with non-probabilistic reliability constraints on the geometrically nonlinear structural behaviour. In order to circumvent numerical difficulties in solving the nested double-loop optimization problem, a performance measure-based approach is employed to transform the constraint on the reliability index into one on the concerned performance. In conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints, the optimization problem is solved by gradient-based mathematical programming methods. Three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method. The obtained optimal solutions show the importance of incorporating various uncertainties in the design problem. Moreover, it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation. [Copyright &y& Elsevier]

Abstract: The topology optimization of load-bearing structural components for reducing attitude control efforts of miniature space vehicles is investigated. Based on the derivation of the cold gas consumption rate of three-axis stabilization actuators, it is pointed out that the attitude control efforts associated with cold gas micro thrusters are closely related to the mass moment inertia of the system. Therefore, the need to restrict the mass moments of inertia of the structural components is highlighted in the design of the load-bearing structural components when the attitude control performance is concerned. The optimal layout design of the space vehicle structure considering attitude control effort is, thus, reformulated as a topology optimization problem for minimum compliance under constraints on mass moments of inertia. Numerical techniques for the optimization problem are discussed. For the case of a single constraint on the mass moment of inertia about a given axis, a design variable updating scheme based on the Karush–Kuhn–Tucker optimality criteria is used to solve the minimization problem. For the problem with multiple constraints, mathematical programming approach is employed to seek the optimum. Numerical examples will be given to demonstrate the validity and applicability of the present problem statement. [Copyright &y& Elsevier]

Abstract: This paper investigates the optimal spatial distribution of single-channel actuation voltage in static structural shape control problem. It is pointed out that single-channel actuation voltage input for shape control applications is of practical importance due to its ease of implementation. The Mindlin plate theory is used in the finite element modeling of laminates with piezoelectric layers. The optimal distribution of the single-channel actuation voltage is formulated as a discrete optimization problem with tri-level design variables, where a least-square function measuring the structural shape error is to be minimized under a constraint on the control effort. Such a problem is then transformed into a continuous one by introducing element-wise artificial design variables defining the topological feature of the control voltage distribution. A power-law function relating the design variables and the applied voltages is proposed to penalize intermediate values of the design variables. Based on the sensitivity analysis, the problem is solved effectively using the MMA algorithm tailored for constrained least-square problems. Numerical examples presented demonstrate the validity of the proposed problem formulation and numerical approaches. [Copyright &y& Elsevier]

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]

Initial thickness imperfections of thin membranes due to manufacturing are inevitable and commonly observed. This paper presents an experimental and numerical investigation on how thickness imperfections affect the wrinkling of stretched membrane structures. With the measured data from a limited number of samples, we model the uncertain deviation of membrane thickness from that of a perfect geometry as a non-probabilistic bounded field with spatial dependency. After representing the bounded field as a linear function of a reduced set of uncertain coefficients by using a series expansion, the effect of thickness imperfections on the membrane wrinkling is investigated. To this end, the upper and lower bounds of out-of-plane displacements for membranes with initial imperfections are evaluated by a mathematical programming. The proposed wrinkling evaluation problem is then solved efficiently by using the Kriging-based optimization algorithm, with validation done by Monte Carlo simulation. The results show that initial thickness imperfections play a critical role in the wrinkling behavior of membranes. [ABSTRACT FROM AUTHOR]

This paper presents a level set-based topology optimization method considering the overhang constraint in additive manufacturing (AM) processes. Though the combination of the topology optimization and AM shows a promising potential and high design flexibility, there are still certain limitations. The overhang constraint is one of the major issues that need to be considered in the design stage. It requires the inclination angles of structural downward-facing surfaces to be larger than a given lower bound, so as to prevent the structure from warping or collapsing during the AM process. We propose a new form of overhang constraint in the level set framework, which is expressed as a single domain integral instead of point-wise constraints. This domain integral form facilitates the detection of overhang constraint violation. The shape derivative of the overhang constraint is derived by using the signed distance property of the level set function. The proposed method is capable of dealing with constraints with different minimum overhang angles. Theoretically, it allows the optimization to proceed from an arbitrary structural layout, without the need to satisfy the overhang constraint in the initial design. Several numerical examples are given to show the validity and effectiveness of the proposed method. It is seen in these examples that the overhang constraint is satisfied mainly by adjusting the local shape of structural members violating the overhang constraint during the optimization process. Thus, the overhang angle constrained optimization can generate similar load paths as in conventional optimal designs in most cases, without significantly worsening the structural stiffness. [ABSTRACT FROM AUTHOR]

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]

Hyperelastic structures usually undergo large deformations and thus may be subject to deformation-dependent contact supports. This paper presents an effective topology optimization methodology for the compliance-minimization design of hyperelastic structures with frictionless contact supports. In the optimization model, the strain-energy function of hyperelastic material is represented by an artificial penalization model, and the contact boundary conditions are modeled with hypothetical nonlinear springs. The additive hyperelasticity technique is employed for circumventing the local buckling instability exhibited by low-density elements. In conjunction with the adjoint variable sensitivity analysis, the nonlinear topology optimization problem is solved by a gradient-based mathematical programming algorithm. Numerical examples are given to show the importance of considering contact supports and to demonstrate the applicability of the proposed method. [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]

Abstract: By introducing a new reduction parameter into the Kreisselmeier–Steihauser (K–S) function, this paper presents a general K–S formulation providing an approximation to the feasible region restricted by active constraints. The approximation is highly accurate even when the aggregation parameter takes a relatively small value. Numerical difficulties, such as high nonlinearity and serious violation of local constraints that may be exhibited by the original K–S function, are thus effectively alleviated. In the considered topology optimization problem, the material volume is to be minimized under local von Mises stress constraints imposed on all the finite elements. An enhanced aggregation algorithm based on the general K–S function, in conjunction with a “removal and re-generation” strategy for selecting the active constraints, is then proposed to treat the stress constraints of the optimization problem. Numerical examples are given to demonstrate the validity of the present algorithm. It is shown that the proposed method can achieve reasonable solutions with a high computational efficiency in handling large-scale stress constrained topology optimization problems. [Copyright &y& Elsevier]

Abstract: This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method. [Copyright &y& Elsevier]

Light-weight cellular materials with periodic repetitive microstructures are widely used in various fields due to their superior mechanical/multi-physical performances. As the microstructural design problem is known to have multiple local minima, most gradient-based topology optimization methods significantly depend on the initial guess of the microstructural geometry, thus requiring the designer's experiences. This paper presents an effective gradient-free framework for periodic microstructure design, which exhibits powerful global searching capabilities and requires no sensitivity information. The proposed framework combines the material-field series-expansion (MFSE) topology representation of periodic microstructures and the sequential Kriging-based optimization algorithm. The MFSE method decouples the topological representation from the finite element discretization, and describes a relatively complex microstructural topology with high-quality boundary description using a greatly reduced number of design variables. Based on the Kriging surrogate model, a solution scheme is suggested to solve material microstructural topology optimization successively in a sequence of sub-optimization problems with self-adaptive design spaces. With the present gradient-free optimization method, high-performance cellular materials that approach the H-S upper bound with porosities from 0.2 to 0.6, or that achieve negative Poisson's ratios of -0.94 in the principal directions for materials with square symmetry, are obtained without prior knowledge of the optimum microstructural topology. Illustration : A gradient-free topological design methods for periodic cellular material microstructures is proposed. This optimization method requires no experienced-based guesses of the initial microstructural topologies and requires no sensitivity information. Microstructure geometries with high-quality boundaries and high performances are obtained with the present method. Unlabelled Image • A gradient-free microstructural topology optimization method for cellular materials is proposed. • This method does not require experience-based guesses of the initial design and the sensitivity information. • The performances of the optimized cellular materials approach the H-S upper bound. • This method provides high-quality microstructural boundary descriptions. [ABSTRACT FROM AUTHOR]