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• An optimum-pursuing method is developed for constrained optimization and reliability-based design optimization problems. • An optimum-pursuing function is proposed from the viewpoint of the optimization theory. • The well-known U function is rigorously derived based on the proposed optimum-pursuing function. • Optimum-pursuing method outperforms alternative methods in terms of both accuracy and efficiency. This paper proposes a new active learning method named as optimum-pursuing method (OPM) from the viewpoint of optimization theory, which aims to provide an effective tool for solving constrained optimization and reliability-based design optimization (RBDO) problems with low computation cost. It uses the cheap Kriging metamodel to replace the expensive physical response. The novelty of the proposed OPM primarily lies in two aspects. First, the OPM utilizes the advantage of the optimization theory rather than sampling technology. By using the augmented Lagrangian approach, it comprehensively considers the objective, constraints, and their relations, thereby automatic identification of important region in the vicinity of the optimum. Second, the accordingly optimum-pursuing function consists of three parts: Kriging mean, Kriging standard deviation, and merit function. Also, the target reliability surface is further considered to enhance the local accuracy of the reliability analysis. The performance of OPM is tested for both deterministic optimization and problems, in which two mathematical and three real-world engineering examples are selected to showcase the feasibility and validity. The results demonstrate that OPM is promising for solving both deterministic optimization and RBDO problems by comparing with the well-known active learning methods. [ABSTRACT FROM AUTHOR]

Dehomogenization techniques are becoming increasingly popular for enhancing lattice designs of compliant mechanical systems with ultra-large resolutions. Their effectiveness hinges on computing a deformed periodic grid that enable to reconstruct fine-scale designs with modulated and oriented patterns. In this paper, we propose an approach for extending dehomogenization methods to laminar fluid systems. We initiate our methodology by asymptotically deriving Darcy's law on a periodically porous medium deformed by a diffeomorphism. Unlike the mechanical context, we reveal that the homogenized permeability matrix depends not solely on local the orientation but also on the local dilation of the deformed periodic medium. This distinction presents one of the several challenges to be tackled when adapting dehomogenization-based topology optimization techniques to porous media. To accommodate existing methodologies, we formulate a simplified "poor man's" homogenized model, which streamlines various aspects, yet still leans on periodic cell problems to estimate the spatially varying permeability matrix. Specifically, we overlook boundary layer effects, we presume periodic grid deformations, and we neglect local dilation, solely considering the relationship with local cell orientations. Subsequently, we present a numerical approach for designing a system that redistributes an input flow across numerous regularly spaced outlets at an output interface. Leveraging the homogenized model, we deduce optimized geometric arrangements of local channel spacing parameters and orientations. We then use established methods to reconstruct grid deformations and fine-scale designs. The fidelity of these reconstructions is then validated through fine-scale simulations. Our observations indicate that while the proposed designs yield satisfactory performance when subjected to the full-scale model, discernible deviations from the homogenized model persist, appealing to future improvements. [ABSTRACT FROM AUTHOR]

This paper presents a minimal mass design approach to the clustered tensegrity structures (CTS). By minimal mass, we mean, for a given clustered tensegrity structure subject to prescribed external loads, the minimal mass is achieved when all the structure members simultaneously fail (buckle or yield). This paper helps to find the lower bound of the required mass of the CTS. Firstly, we introduce the connectivity and clustering definitions through respective matrices and derive the nonlinear clustered tensegrity statics equation in terms of the nodal vector of the structure. Since the mass of each member is in a one-to-one correspondence to its force density (force by unit length), the static equilibrium equations of the CTS with respect to the force density and force vectors are also given. Then, we formulated the mass and gravity of hollow bars and strings subject to buckling and yielding conditions. Finally, a nonlinear optimization algorithm is presented to compute the minimal mass of any CTS with any given external force and topology. We also demonstrate that the traditional tensegrity structure (TTS) is a particular case of the CTS by defining the cluster matrix as an identity matrix. At last, numerical examples are given to validate the CTS minimal mass design approach. The method proposed by this research can be used for the lightweight design of tensegrity, truss, cable nets, membrane structures, etc. • A method to calculate the least necessary mass of any clustered tensegrity. • The traditional tensegrity is a particular case of clustered tensegrity. • The clustering strategy can reduce the number of strings, but the payment is mass. • The method allows hollow/solid bar tensegrity design with/without gravity. [ABSTRACT FROM AUTHOR]

The present paper focuses on solving partial differential equations in domains exhibiting symmetries and periodic boundary conditions for the purpose of homogenization. We show in a systematic manner how the symmetry can be exploited to significantly reduce the complexity of the problem and the computational burden. This is especially relevant in inverse problems, when one needs to solve the partial differential equation (the primal problem) many times in an optimization algorithm. The main motivation of our study is inverse homogenization used to design architected composite materials with novel properties which are being fabricated at ever increasing rates thanks to recent advances in additive manufacturing. For example, one may optimize the morphology of a two-phase composite unit cell to achieve isotropic homogenized properties with maximal bulk modulus and minimal Poisson ratio. Typically, the isotropy is enforced by applying constraints to the optimization problem. However, in two dimensions, one can alternatively optimize the morphology of an equilateral triangle and then rotate and reflect the triangle to form a space filling D 3 symmetric hexagonal unit cell that necessarily exhibits isotropic homogenized properties. One can further use this D 3 symmetry to reduce the computational expense by performing the “unit strain” periodic boundary condition simulations on the single triangle symmetry sector rather than the six fold larger hexagon. In this paper we use group representation theory to derive the necessary periodic boundary conditions on the symmetry sectors of unit cells. The developments are done in a general setting, and specialized to the two-dimensional dihedral symmetries of the abelian D 2 , i.e. orthotropic, square unit cell and nonabelian D 3 , i.e. trigonal, hexagon unit cell. We then demonstrate how this theory can be applied by evaluating the homogenized properties of a two-phase planar composite over the triangle symmetry sector of a D 3 symmetric hexagonal unit cell. [ABSTRACT FROM AUTHOR]

Calibration experiment design optimization (CEDO) seeks to identify the optimal values of experimental inputs in order to maximize the obtained information within testing budget constraints. Current CEDO methods only consider observation errors and focus on problems with low-dimensional response variables. This paper proposes a new robust CEDO model for Bayesian calibration with field response and incorporates various sources of model uncertainty such as surrogate model uncertainty, numerical discretization error, and model discrepancy, into CEDO. Since sampling-based methods to estimate the posterior distributions and thus compute the objective function (information gain) are computationally expensive, this paper develops an analytical method to estimate the objective function while accounting for model uncertainty during this process. Based on the evaluation of objective function, a global sensitivity analysis (GSA)-based method is proposed to check the quality of surrogate models used in CEDO, guide the training and improvement of the surrogate models, and thus reduce the effect of surrogate model uncertainty on CEDO. The optimal experimental input settings are obtained using the efficient global optimization (EGO) method. A nonlinear analytical example and a heat transfer example are used to illustrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a robust topology optimization methodology for structures with imprecise probability uncertainty. In this paper, the imprecise probability uncertainties are treated with an interval random model, in which the probability variables are used to model the uncertain parameters and some distribution parameters of probability variables are expressed as interval variables instead of a precise value. In the presented methodology, the deterministic topology optimization techniques and the hybrid stochastic interval perturbation method (HSIPM) are combined to obtain robust topology designs for structures with interval random parameters. The exploitation of HSIPM transforms the problem of topology optimization with interval random parameters into an augmented deterministic topology optimization problem. This provides a computationally cheap alternative to Monte Carlo-based optimization algorithms. Several numerical examples are presented to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract To reduce the computational cost of solving engineering design optimization problems with both expensive objective and constraints, a novel filter-based adaptive Kriging method notated as FLT-AKM is proposed in this paper. In FLT-AKM, a probability of constrained improvement (PCI) criterion is developed based on the notion of filter to sequentially generate new samples for updating Kriging metamodels of objective and constraints. At each iteration, an infill sample point is allocated at the position where the PCI is maximized to achieve potential improvement in optimality and feasibility. And the Kriging metamodels are consecutively updated by the newly-added infill sample points, which leads the FLT-AKM search to rapidly converge to the global optimum. The performance of the proposed FLT-AKM method is tested on a number of numerical benchmark problems via comparing with several widely-used metamodel-based constrained optimization methods. The comparison results indicate that FLT-AKM generally outperforms the competitors in terms of global convergence and efficiency performance. Finally, FLT-AKM is successfully applied to an all-electric GEO satellite MDO problem. The optimization results show that FLT-AKM is able to find a better feasible design with fewer computational budgets compared with our previous study, which demonstrates the effectiveness and practicality of the proposed FLT-AKM method for solving real-world expensive black-box engineering design optimization problems. [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 Reliability-based design optimization (RBDO) is a powerful tool to handle the influence of various uncertainties during optimization. However, unbearable computation cost is one of the largest barriers for its application, especially for the finite element method (FEM)-based RBDO. In this paper, an efficient and accurate RBDO framework is established based on isogeometric analysis (IGA) for complex engineering problems. Furthermore, an enhanced step length adjustment (ESLA) iterative algorithm and a second-order reliability method-based stepped-up sequential optimization and reliability assessment approach (SSORA-SORM) are proposed to boost the efficiency of RBDO. According to the situation of iteration process, the step length of search the most probable target point can be adaptively updated by the proposed criterion to improve the robustness in ESLA. In the proposed framework, the analytical first-order sensitivity is derived based on IGA in the optimization process to substitute the time-consuming finite difference method. The robustness, accuracy and efficiency of proposed methods are verified via several numerical benchmarks. Besides, three complex IGA-based examples demonstrate that the proposed method is able to save much computational cost without losing accuracy, which is inherently suitable for the RBDO of complex engineering problems. [ABSTRACT FROM AUTHOR]

Abstract The integrity of engineering structures is often compromised by embedded surfaces that result from incomplete bonding during the manufacturing process, or initiation of damage from fatigue or impact processes. Examples include delaminations in composite materials, incomplete weld bonds when joining two components, and internal crack planes that may form when a structure is damaged. In many cases the areas of the structure in question may not be easily accessible, thus precluding the direct assessment of structural integrity. In this paper, we present a gradient-based, partial differential equation (PDE)-constrained optimization approach for solving the inverse problem of interface detection in the context of steady-state dynamics. An objective function is defined that represents the difference between the model predictions of structural response at a set of spatial locations, and the experimentally measured responses. One of the contributions of our work is a novel representation of the design variables using a density field that takes values in the range [ 0 , 1 ] and raised to an integer exponent that promotes solutions to be near the extrema of the range. The density field is combined with the penalty method for enforcing a zero gap condition and realizing partially bonded surfaces. The use of the penalty method with a density field representation leads to objective functions that are continuously differentiable with respect to the unknown parameters, enabling the use of efficient gradient-based optimization algorithms. Numerical examples of delaminated plates are presented to demonstrate the feasibility of the approach. Highlights • We present a method for identifying partially connected surfaces. • Our formulation leads to a differentiable objective function. • We use a density field representation of the design variables. • Our method can be applied to arbitrary geometries. [ABSTRACT FROM AUTHOR]

Abstract Metamaterials with hyperbolic equal-frequency contours possess unique properties, which offer novel means in elastic wave manipulation. The existing designs are either resonant-structure based that have large damping loss or have a relatively narrow working frequency range. In this paper, a topology optimization approach is developed to design non-resonant hyperbolic elastic metamaterials over a broad frequency range. Based on the physical nature of hyperbolic metamaterials, a two-step topology optimization design scheme is formulated and solved by a gradient-based optimizer. By engineering the dispersion band structure, non-resonant hyperbolic metamaterials have been successfully designed. Novel features such as negative refraction, wave partial focusing and super-resolution imaging have been numerically demonstrated. Highlights • A topology optimization scheme is proposed for hyperbolic elastic metamaterial design. • Structural integrity is ensured by imposing constraint on static effective modulus. • Novel wave features are demonstrated over broad frequency range. • Directional and polarization-dependent transmission has been achieved. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an isogeometric analysis (IGA) based design method to address the stress-constrained topology optimization problem of plane stress and bending of thin plates. Based on the popular SIMP model, the integrated framework of geometry modeling, structural stress analysis and optimization is established. Owing to the geometry exactness and high-order continuity between elements, the IGA improves the computational accuracy of stress, and thus enhances the credibility of optimum design. Meanwhile, the obvious zigzag boundaries are avoided in the optimized results, and the stress function of IGA maintains the continuity for a relatively coarse discretization. Moreover, the IGA-SIMP method can easily meet the requirement of C 1 -continuity for the Kirchhoff plate formulations, which also facilitates the stress analysis and sensitivity calculation. To overcome the convergence difficulty of highly nonlinear stress aggregation constraint, two STM (stability transformation method)-based stabilization schemes combining with the P-norm function for global stress constraint are developed to achieve the stable iterations and acceptable designs. Finally, representative examples illustrate the effectiveness and convenience of the proposed approach. It is indicated that the IGA-SIMP method shows superior performance for solution accuracy and efficiency, and the local stress level is well controlled. [ABSTRACT FROM AUTHOR]

The application of isogeometric analysis requires a full parametrization map of the physical domain. This map is suitable for the analysis only if it is injective. In this paper we give a necessary condition for the injectivity of a B-spline biquadratic map. Moreover, sufficient injectivity conditions, which are easy to check in the practice, are introduced. A method for parametrizing planar regions using a biquadratic map is also proposed. The method is simpler than others reported in the literature, since it is based on the solution of an optimization problem without restrictions and only requires the construction of a quadrilateral mesh with optimal geometric properties. Finally we show that the proposed method can be successfully used to obtain high quality parametrizations of simply connected planar regions with irregular boundary. [ABSTRACT FROM AUTHOR]

Symmetry and repetitive patterns are very common in both natural and artificial structures. Design with symmetry and pattern repetition (SPR) can be very useful in practical engineering. In this paper, we discuss a new optimal structural design method considering SPR constraints. The basic idea consists in combining a parameterized level-set method together with radial basis function (RBF) interpolation strategy, which allows for decoupling RBF knots from the finite element (FE) analysis mesh. The merits of the decoupled manner are that RBF knots can be mapped independently on regular or irregular design domain, and SPR constraints can be imposed by simply building a bijective relationship between RBF knots regardless of analysis mesh. Furthermore, due to parameterization of the level-set function, there are fewer numerical manipulations that need to be implemented than the conventional level-set method during the optimization process. Several numerical examples are presented to show feasibility and effectiveness of the proposed method. [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 introduces a new optimization framework to multiscale structures with unknown-but-bounded (UBB) uncertainties, where the optimal design is applied at two scales: the macroscale, i.e., the structural level, where the non-probabilistic reliability-based topology optimization (NRBTO) is conducted, and the microscale, i.e., the material level, where the non-probabilistic reliability-based design optimization (NRBDO) for truss-like cell dimensions is achieved. By combination of the homogenization method and interval mathematics, rules of macro–micro stiffness equivalence as well as interval bounds of macro displacement responses under specific micro structural settings can be first confirmed. For safety reasons, novel reliability definitions are respectively given for the double-level design, in which the offset factor with good differential properties is applied for the NRBTO and the matching factor derived from the improved rule of interval sorting is utilized for the NRBDO. To circumvent problems of large-scale variable updating in two-level optimization procedures, the adjoint-based vector in macro layout design as well as the particle swarm optimization (PSO) algorithm in micro parametric design are further discussed. The usage, effectiveness, and the superiority of the developed methodology are eventually demonstrated by two 3D engineering applications. [ABSTRACT FROM AUTHOR]

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

Low density cellular materials may offer excellent mechanical properties and find wide applicability in lightweight design and infill structures for additive manufacturing, yet currently existing material structures are still far away from their theoretical limit in terms of compressive strength. To explore the existing potential, this paper presents a topology optimization framework for designing periodic cellular materials with maximized strength under compressive loading. Under this condition, the limiting factor of strength is the failure mechanism of buckling instability in the microstructure. In order to predict microstructural buckling, a simplified model based on homogenization theory, a linearized stability criterion and Floquet–Bloch theory is employed. Subsequently, a gradient-based topology optimization problem is formulated to maximize the buckling strength of the most critical failure mode. The framework is utilized to optimize square, triangular and hexagonal microstructures for three different macroscopic load conditions including biaxial, uniaxial and shear loading, and performance assessments are conducted by computation of associated failure surfaces in macroscopic stress space. In all cases, the optimized designs turn out to be first-order hierarchical type microstructures which offer major improvements of strength compared to the initial zero-order designs, however, the gains come at the cost of reductions in stiffness. Furthermore, it is illustrated how imposing geometric symmetry constraints can be exploited to control the shape of the failure surfaces. [ABSTRACT FROM AUTHOR]

This paper aims to present an elegant way to design multimodal piezo transducers, i.e., piezoelectric sensors/actuators that succeed in filtering a set of desired eigenmodes among those belonging to a bigger set prescribed beforehand. That can be efficiently done by regarding the design problem as one of the topology optimization, where the design variable taking on the values either −1 or 1 physically represents the polarity of the electrodes. It can be analytically proved that this optimization problem has classical solution, and indeed it is unique. Motivated by fabrication issues happening at the micro scale, it is also considered the robust version of that problem, that is, the design of such devices whenever the feature size of both polarity phases are underneath limited. [ABSTRACT FROM AUTHOR]

The robustness and efficiency of performance measure approach (PMA) depend on the reliability loop in reliability-based design optimization (RBDO). For the reliability loop in the PMA using the minimum performance target point (MPTP) search, existing approaches can obtain stable results but may converge to inaccurate results, and higher computational efforts are required to achieve the optimum results for highly nonlinear problems. In this paper, a hybrid descent mean value (HDMV) approach is proposed based on a novel merit function, which is applied to combine the MPTP search formulas of the descent mean value (DMV) and advanced mean value (AMV). The merit function is used to adaptively control the numerical instability of the inverse reliability analysis for RBDO-based PMA. The accuracy, robustness and efficiency of the proposed DMV and HDMV methods are compared with existing methods through four nonlinear performance functions, two structural RBDO problems and a complex aircraft panel problem. The results illustrate that the DMV and HDMV methods are more robust, efficient and accurate than existing reliability methods. For the aircraft panel problem, a simultaneous buckling pattern is finally achieved by the proposed methods with better performance in terms of both convergence rate and computational efficiency. [ABSTRACT FROM AUTHOR]

This paper investigates a new integrated optimization method for laminated composite structures with piezoelectric patch actuators. Three kinds of design variables, i.e. the actuators’ locations, electrical voltages applied on piezoelectric patches and host structural pseudo-densities, are simultaneously optimized to improve the overall deformation precision. To freely place piezoelectric actuator on the host structure surface regardless of coincident finite element in classic lamination theory (CLT), the multi-point constraints (MPC) method is used here to simulate the perfect bonding between actuator layer and host layer, which inherits the advantages of avoidance of remeshing process analytical sensitivity as well as efficient lamination description during the movement of actuators in optimization iterations. For applying to spatially complex shape control, a modified shape error function based on the relative error between computed and desired surfaces is used. The finite circle method (FCM) is implemented to prevent the overlaps among the actuators and those between actuators and boundaries of the global design domain. Finally, numerical examples with plane or curved host structure are tested and discussed to demonstrate the validity and efficiency of this method. [ABSTRACT FROM AUTHOR]

In this paper, a novel topology optimization method based on discrete cosine transform (DCT) and density interpolation is proposed for layout designs of 2D and 3D continua. As one of the most frequently used transforms in digital image compression, the DCT may significantly reduce the number of design variables in density-based topology optimization, and can hereby improve the efficiency of solving the topology optimization problems to a great extent. This way the DCT compression based density method (DCDM) could be quite attractive in the topology optimization of large-scale engineering structures where a huge number of design variables may present. Effectiveness and efficiency of the proposed method is demonstrated with several 2D and 3D examples including both mechanical and heat conduction problems. Through these examples, some interesting features of DCDM are revealed and discussed. Since high frequency components are inherently filtered in DCDM, there is no need to introduce additional density filter or sensitivity filter in the present model. It is shown by numerical examples that there is no sharp corners present in the final optimized layout obtained by DCDM, which is beneficial when considering the stress of structures. [ABSTRACT FROM AUTHOR]

The conventional topology optimization considering uncertainties is based on the probabilistic model. This paper proposes a new reliability-based topology optimization (RBTO) approach for the uncertain structural design based on the fuzzy set model, which can be constructed by expert opinions. The formulation of the proposed RBTO is established as a volume minimization problem under constraints established within the possibility context, termed as failure possibility constraints. The computation for the failure possibility constraints in each optimization loop is a double-loop nested problem. Accordingly, the proposed RBTO is a triple-loop nested problem, which is computationally expensive. To improve the computational efficiency, the target performance approach is introduced to the failure possibility constraints, and thus the triple-loop nested problem is reduced to a double-loop nested problem. To further improve the computational efficiency, the interval perturbation method (IPM) is used to estimate the constraints, thereby the calculation for the constraints in each optimization loop becomes deterministic, and the double-loop nested problem is reduced to an approximate single-loop problem. Numerical results on two examples are presented to verify the effectiveness and feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper proposes a level set method to solve minimum stress and stress-constrained shape and topology optimization problems. The method solves a sub-optimization problem every iteration to obtain optimal boundary velocities. A p -norm stress functional is used to aggregate stresses in a single constraint. The shape sensitivity function is derived and a computational procedure based on a least squares interpolation approach is devised in order to compute sensitivities at the boundaries. Adaptive constraint scaling is used to enforce exact control of stress limits. Numerical results show that the method is able to solve the problem efficiently for single and multiple load cases obtaining solutions with smooth boundaries. [ABSTRACT FROM AUTHOR]

Multi-material level set topology optimization methods conventionally rely on the overlap of multiple level set functions to realize the multi-material structural representation. However, this representation may produce redundant material phases and the signed distance information is no longer available within the individual material regions. To fix these issues, a new multi-material level set topology optimization method is proposed in this paper, where m level set functions represent m material phases plus the void, and the signed distance information is straightforwardly available in each material phase. To be specific, each level set function corresponds to a material phase and the overlapping areas are filled with an artificial material type, the property of which is weaker than any of the involved material types. This artificial material plays the role of penalizing the overlap areas to vanish, which is different from the close-to-zero property material type for the void. Hence, the optimization process starts with any multi-level set interpolated input and the overlapping areas will gradually vanish. Based on this new method, we have successfully realized the component length scale control on multi-material structures and innovatively, an approach has been proposed to realize the uniform component thickness control without need of pre-specifying the thickness target. [ABSTRACT FROM AUTHOR]

The optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable, or even lead to catastrophic failures. A two-level optimization formulation is often used in the Robust Topology Optimization (RTO) under uncertain loads. Various approaches have been reported to identify the critical loads associated with the worst structure responses. Because Convex Model approaches apply convex approximations to the original non-convex model at the lower level, the optimal designs obtained by these methods are greatly dependent on the quality of the approximation. In this paper, a new formulation based on the Wolfe duality for the RTO problems with multiple independent unknown-but-bounded loads is proposed. Following the two-level formulation, the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the Wolfe duality. Both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non-convexity associated with the lower level formulation. Numerical examples are also presented to demonstrate the proposed approach. [ABSTRACT FROM AUTHOR]

The present paper describes a method to enhance the capability of, or to broaden the scope of computational mechanics by using deep learning, which is one of the machine learning methods and is based on the artificial neural network. The method utilizes deep learning to extract rules inherent in a computational mechanics application, which usually are implicit and sometimes too complicated to grasp from the large amount of available data A new method of numerical quadrature for the FEM stiffness matrices is developed by using the proposed method, where a kind of optimized quadrature rule superior in accuracy to the standard Gauss–Legendre quadrature is obtained on the element-by-element basis. The detailed formulation of the proposed method is given with the sample application above, and an acceleration technique for the proposed method is discussed [ABSTRACT FROM AUTHOR]

A new non-probabilistic reliability-based topology optimization (NRBTO) method for detailed design of continuum structures is proposed in this paper, in which the unknown but bounded (UBB) uncertainties existing in material and external loads are considered simultaneously. Based on the interval set-theoretic model, traditional displacement constraint is transformed into the non-probabilistic reliability format, in which a novel measuring index, namely, the optimization feature distance is defined by a combination of the area-ratio ideas. Thus, the optimal layout issue is converted by the lightweight design problem with displacement reliability constraints. Moreover, the interval parametric vertex approach and the adjoint vector method are also employed to obtain the sensitivity information between the reliability measurement and design variables. By utilizing the method of moving asymptotes (MMA), the optimization problem that is investigated can be solved iteratively. Two engineering examples are eventually presented to demonstrate the validity and effectiveness of the methodology that has been developed, and numerical results show that considering the UBB uncertainty effects during the whole procedure of topology optimization may have a significant influence on the final structural configurations. [ABSTRACT FROM AUTHOR]

Feature-driven topology optimization has been extensively studied in the past decade, and the majority of the works have treated it as the multi-component/void layout optimization problem. A major limitation of this treatment is that the number of components/voids should be defined in advance. To overcome this limitation, this paper presents a novel void feature control method, through which the specified void feature (in any geometric form) could be well-contained by each interior void in the finalized topology design, regardless of the quantity. Numerical stability of this method is discussed and an auxiliary algorithm has been developed to enhance it. Other than that, the proposed method also serves the purpose of void length scale control, which is also a hot issue in the topology optimization field. To be specific, by tailoring the specified void feature size, the void length scale is guaranteed to be larger than that. To the authors’ knowledge, the void length scale control was rarely studied under the level set framework. Through a few numerical case studies, it is proven that the void feature control method is effective while only limited compromise of the structural performance has been observed. [ABSTRACT FROM AUTHOR]

In topology optimization design, specific requirements over the complexity of component geometry often need to be satisfied for practical purposes. In this paper, with the introduction of a novel complexity control measure built on the structural skeletons of solid and void regions, an explicit structural complexity control approach for the topology optimization of continuum structures is proposed under the framework of level-set representation. The use of skeletons enables us to control the minimum sizes of structural holes so as to meet the manufacturing demands. Through numerical comparison with the cases that have been investigated implicitly, we show that the proposed approach is an effective tool to control the structural complexity in an explicit way. [ABSTRACT FROM AUTHOR]

In this paper, a general and efficient method for constructing LHDs possessing favorable characteristics of orthogonality and better computational efficiency is first proposed. Next, a sequentially enlarging strategy is derived based on the initial design. During the enlarging process, optimization of space-filling property is carried out, and a near-orthogonal space-filling augmented LHD is subsequently obtained. The enlarging process is repeated iteratively until the desired accuracy is achieved. Numerical examples and two mechanical applications are utilized in order to confirm and emphasize the effectiveness of this proposed method. [ABSTRACT FROM AUTHOR]

This paper applies variational data assimilation to inundation problems governed by the shallow water equations with wetting and drying. The objective of the assimilation is to recover an unknown time-varying wave profile at an open ocean boundary from inundation observations. This problem is solved with derivative-based optimisation and an adjoint wetting and drying scheme to efficiently compute sensitivity information. The capabilities of this approach are demonstrated on an idealised sloping beach setup in which the profile of an incoming wave is reconstructed from wet/dry interface observations. The method is robust to noise in the observations if a regularisation term is added to the optimisation objective. Finally, the method is applied to a laboratory experiment of the Hokkaido-Nansei-Oki tsunami, where the wave profile is reconstructed with a relative L ∞ error of less than 1 %. [ABSTRACT FROM AUTHOR]

Three dimensional (3D) topology optimization problems always involve huge numbers of Degrees of Freedom (DOFs) in finite element analysis (FEA) and design variables in numerical optimization, respectively. This will inevitably lead to large computational efforts in the solution process. In the present paper, an efficient and explicit topology optimization approach which can reduce not only the number of design variables but also the number of degrees of freedom in FEA is proposed based on the Moving Morphable Voids (MMVs) solution framework. This is achieved by introducing a set of geometry parameters (e.g., control points of B-spline surfaces) to describe the boundary of a structure explicitly and removing the unnecessary DOFs from the FE model at every step of numerical optimization. Numerical examples demonstrate that the proposed approach does can overcome the bottleneck problems associated with a 3D topology optimization problem in a straightforward way and enhance the solution efficiency significantly. [ABSTRACT FROM AUTHOR]

Starling murmuration optimizer is a newly well-developed swarm intelligence algorithm inspired by the behavior of starlings during stunning murmuration and has performed competitively with other well-known methods. However, starling murmuration optimizer faces the issues of poor search capability, iterative stagnation, and low convergence accuracy when dealing with complex engineering optimization and high-dimensional problems. To ameliorate these deficiencies, an efficient hybrid starling murmuration optimizer that combines dynamic opposition, Taylor-based optimal neighborhood strategy, and crossover operator is developed in this paper. Firstly, introducing the Taylor-based optimal neighborhood strategy ensures a more careful and reasonable reliance on optimal individuals, thus reducing the possibility of falling into local optima. Meanwhile, dynamic opposition successfully balances the exploration and development phases by incorporating the ideas of quasi- and reflective opposition. In addition, the exploration-development capability is further enhanced by introducing the crossover operator. To ensure the adaptivity of the crossover factor, 12 different inertia weights are explored for crossover factors, and the sigmoidal decreasing inertia weights achieve optimality. Finally, the excellent performance of the proposed algorithm is verified by comparing it with the improved algorithm and the state-of-the-art search algorithms on 26 benchmark functions and the CEC2020 test suite, respectively. In addition, the proposed algorithm is used to optimize ten practical engineering optimization problems. Experimental results demonstrate that the proposed algorithm has strong competitiveness in this algorithm regarding computational power and convergence accuracy. Finally, the proposed algorithm is applied to two truss topology optimization designs. The experimental results demonstrate the applicability and potential of the proposed algorithm in practical applications. The proposed algorithm is a competitive optimization algorithm for solving optimization problems. [ABSTRACT FROM AUTHOR]

For multilayer structures, interfacial failure is one of the most important elements related to device reliability. For cohesive zone modeling, traction–separation relations represent the adhesive interactions across interfaces. However, existing theoretical models do not currently capture traction–separation relations that have been extracted using direct methods, particularly under mixed-mode conditions. Given the complexity of the problem, models derived from the neural network approach are attractive. Although they can be trained to model data along the loading paths taken in a particular set of mixed-mode fracture experiments, they may fail to obey physical laws for paths not covered by the training data sets. In this paper, a thermodynamically consistent neural network (TCNN) approach is established to model the constitutive behavior of interfaces when faced with sparse training data sets. Accordingly, four conditions are examined and implemented here: (i), positive energy dissipation; (ii), maximum energy 10 dissipation; (iii), conservation of the J-integral; (iv), dependence of interfacial toughness on the fracture mode-mix. These conditions are treated as constraints and are implemented as such in the loss function. The feasibility of this approach is demonstrated by comparing the modeling results with a range of physical constraints and several different input data sets. Moreover, a Bayesian optimization algorithm is then adopted to optimize the weight factors associated with each of the constraints in order to overcome convergence issues that can arise when multiple constraints are present. The resultant numerical implementation of the ideas presented here produced well-behaved, mixed-mode traction–separation​ surfaces that maintained the fidelity of the experimental data that was provided as input. Some guidance is also provided on the desirable features of input data sets. The proposed approach heralds a new autonomous, point-to-point constitutive modeling concept for interface mechanics. [ABSTRACT FROM AUTHOR]

This paper proposes a self-support topology optimization method that considers distortion to improve the manufacturability of additive manufacturing. First, a self-support constraint is proposed that combines an overhang angle constraint with an adjustable degree of the downward convex shapes and a thermal constraint for heat dissipation in the building process. Next, we introduce a mechanical model based on the inherent strain method in the building process and propose a constraint that can suppress distortion. An optimization problem is formulated to satisfy all constraints, and an optimization algorithm based on level-set-based topology optimization is constructed. Finally, two- and three-dimensional optimization examples are presented to validate the effectiveness of the proposed topology optimization method. • A new self-support constraint based on the level-set-based topology optimization is proposed. • A distortion constraint in additive manufacturing (AM) based on the inherent strain method is formulated. • Self-support topology optimization considering distortion for metal AM is proposed. • Optimized designs satisfy each constraint without much loss of structural performance. [ABSTRACT FROM AUTHOR]

Abstract: This paper focuses on the construction of statically admissible stress fields (SA-fields) for a posteriori error estimation. In the context of verification, the recovery of such fields enables to provide strict upper bounds of the energy norm of the discretization error between the known finite element solution and the unavailable exact solution. The reconstruction is a difficult and decisive step insofar as the effectiveness of the estimator strongly depends on the quality of the SA-fields. This paper examines the strong prolongation hypothesis, which is the starting point of the Element Equilibration Technique (EET). We manage to characterize the full space of SA-fields satisfying the prolongation equation so that optimizing inside this space is possible. The computation exploits topological properties of the mesh so that implementation is easy and costs remain controlled. In this paper, we describe the new technique in details and compare it to the classical EET and to the flux-free technique for different 2D mechanical problems. The method is explained on first degree triangular elements, but we show how extensions to different elements and to 3D are straightforward. [Copyright &y& Elsevier]

With the availability of multi-material additive manufacturing, topology optimization of a multi-material structure and the selection of candidate materials become increasingly important. This paper aims to develop a new multi-material topology optimization algorithm and propose the guidelines for the selection of candidate materials from the database. The multi-material design variables and their inequality relationship are built on volume fractions of multiple materials within each element. The multi-material design variables are relaxed and multiple floating projection constraints simulate their discrete constraints by pushing design variables towards 0 or 1. Meanwhile, their inequality relationship is enforced by their variation limits. The proposed multi-material topology optimization algorithm can be applied to the compliance minimization problem constrained by a single mass or multiple volumes, as demonstrated in numerical examples. This paper mainly focuses on a single mass constraint, and 2D and 3D numerical examples systematically demonstrate that an optimized design under a mass constraint achieves a lower compliance when more materials appear in the final design simultaneously. Furthermore, we establish an approach to predict the inclusion or exclusion of a material from the final design, and propose the conditions for the co-existence of candidate materials, which guide users to select candidate materials from the database. • A new multi-material topology optimization algorithm is proposed. • The algorithm can be directly applied to a single mass constraint or multiple volume constraints. • The results systematically demonstrate the advantage of the multi-material design. • The approach to predict the occurrence of candidate materials in the final design is developed. • The approach can guide the users to select materials for multi-material additive manufacturing. [ABSTRACT FROM AUTHOR]

The signed distance function (SDF) gives the shortest distance from a given point to the boundary, and the sign indicates whether this point is inside or outside the closed boundary or enclosed region. The SDF property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance-based interpolation of different material properties. In conventional level set methods, a common way of achieving a level set function with the signed distance property is to periodically implement the so-called reinitialization scheme by solving an additional Hamilton–Jacobi partial differential equation. However, such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended, which may shift the optimization result and bring convergence issues. In this paper, a double-well potential functional is employed for distance regularization inside the topology optimization loop, which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain. The radial basis function (RBF) based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance-regularized topology optimization method in handling problems with non-convex objective functions and multiple constraints. The flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process. Both 2D and 3D benchmark examples are employed to demonstrate the validity of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies the characteristics and applicability of the CutFEM approach of Burman et al. (2014) as the core of a robust topology optimization framework for 3D laminar incompressible flow and species transport problems at low Reynolds numbers ( R e < 200 ). CutFEM is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques. In this study, the geometry of the fluid domain is described by an explicit level set method, where the parameters of a discretized level set function are defined as functions of the optimization variables. The fluid behavior is modeled by the incompressible Navier–Stokes equations. Species transport is modeled by an advection–diffusion equation. The governing equations are discretized in space by a generalized extended finite element method. Face-oriented ghost-penalty terms are added for stability reasons and to improve the conditioning of the system. The boundary conditions are enforced weakly via Nitsche’s method. The emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem. An auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure. Numerical results for 3D, steady-state and transient problems demonstrate that the CutFEM analyses are sufficiently accurate, and the optimized designs agree well with results from prior studies solved in 2D or by density approaches. [ABSTRACT FROM AUTHOR]

This paper aims to utilize the Isogeometric Analysis (IGA) for the level set structural topology optimization. The level set function is parametrized using the Non-Uniform Rational B-Spline (NURBS) basis functions in a higher dimension. The same basis functions are employed for approximating the unknown deformations, geometry modeling of the design domain and the level set function. In this research, three optimization problems including minimization of the mean compliance considering a certain amount of material, minimization of weight with avoiding local stress concentration as well as minimization of weight and strain energy under local stress constraints are dealt with. The sensitivity analyses for the optimization problems are carried out to obtain velocity functions on the boundaries. In order to move the boundaries towards optimum the Hamilton–Jacobi (H–J) equation is solved using the forward Euler scheme. In order to illustrate the performance of the method to obtain reasonable results with smooth boundaries, several numerical examples are presented and compared with well-known problems in literature. [ABSTRACT FROM AUTHOR]

This paper proposes a level set-based robust topology optimization (RTO) method for computational design of metamaterials under hybrid uncertainties, e.g. auxetics with negative Poisson’s ratio, where the Young’s modulus of the solid is described as a random variable while the Poisson’s ratio is regarded as an interval variable. Firstly, the robust objective function is formulated by a combination of interval mean and interval variance of the deterministic objective function. Secondly, the interval mean and interval variance are computed by a hybrid uncertain analysis approach, termed as Polynomial Chaos-Chebyshev Interval (PCCI) method. Thirdly, the design sensitivities of the robust objective function are obtained after the implementation of the PCCI method. Finally, a powerful parametric level set method (PLSM) in conjunction with the numerical homogenization method is applied to achieve the robust topological design for the auxetic microstructure. Several numerical cases are used to demonstrate the effectiveness of the proposed method for the robust topology optimization problems. This method is non-intrusive and general, and can be easily extended to a range of design problems of micro-structured metamaterials. [ABSTRACT FROM AUTHOR]

This paper is devoted to acquiring strict bounds on discretization error as well as quantities of interest for contact problems. On the one hand, there are two key steps in estimating the discretization error: the first is the proper construction of approximate contact forces and the second is the utilization of the strict upper bounding property inherent in the constitutive relation error (CRE). On the other hand, upper and lower bounds on quantities of interest are derived through an optimization procedure. Academic examples are performed to verify the proposed bounds. [ABSTRACT FROM AUTHOR]

This paper presents isogeometric topology optimization (ITO) for periodic lattice materials, where non-uniform rational B-spline (NURBS) basis functions of CAD models are directly used in the finite element analysis to improve computational accuracy and efficiency. Two TO schemes that use asymptotic homogenization (AH) for the calculation of the mechanical properties are proposed for lattice materials with uniform and graded relative density respectively. To accelerate ITO for graded lattice materials, the mechanical properties are expressed as a function of the relative density of the unit cell, a step that avoids their iterative calculations during ITO. Three benchmark examples are presented to validate the proposed scheme with results that show tangible advantages, such as reduced computational time and faster convergence, of ITO over conventional TO. [ABSTRACT FROM AUTHOR]

This paper presents an alternative topology optimization method for bounded acoustic problems that uses the hybrid finite element-wave based method (FE-WBM). The conventional method for the topology optimization of bounded acoustic problems is based on the finite element method (FEM), which is limited to low frequency applications due to considerable computational efforts. To this end, we propose a gradient-based topology optimization method that uses the hybrid FE-WBM whereby the entire domain of a problem is partitioned into design and non-design domains. In this respect, the FEM is used as a design domain of topology optimization, and the WBM is used as a non-design domain to increase computational efficiency. The adjoint variable method based on the hybrid FE-WBM is also proposed as a means of computing design sensitivities. Numerical examples are presented to demonstrate the effectiveness of the proposed method. We compare the optimized design obtained from the proposed method to that obtained from the conventional method in terms of objective function values, optimized topologies and computational efficiency. The optimization results show that the proposed method can perform more efficient topology optimization than conventional method and can thus be applied to much higher frequency applications that conventional method takes considerable computation time to manage. [ABSTRACT FROM AUTHOR]

This paper presents a design methodology based on a combination of isogeometric analysis (IGA), level set and point wise density mapping techniques for topology optimization of piezoelectric/flexoelectric materials. The fourth order partial differential equations (PDEs) of flexoelectricity, which require at least C 1 continuous approximations, are discretized using Non-Uniform Rational B-spline (NURBS). The point wise density mapping technique with consistent derivatives is directly used in the weak form of the governing equations. The boundary of the design domain is implicitly represented by a level set function. The accuracy of the IGA model is confirmed through numerical examples including a cantilever beam under a point load and a truncated pyramid under compression with different electrical boundary conditions. Finally, we provide numerical examples demonstrating the significant enhancement in electromechanical coupling coefficient that can be obtained using topology optimization. [ABSTRACT FROM AUTHOR]

Chameleon swarm algorithm (CSA) is a newly proposed swarm intelligence algorithm inspired by the chameleon's foraging strategies of tracking, searching and attacking targets, and has shown well competitive performance with other state-of-the-art algorithms. Interestingly, CSA mathematically models and implements the steps of chameleon's unique food-seeking behavior. Nevertheless, the original CSA suffers from the challenges of insufficient exploitation ability, ease of falling into local optima, and low convergence accuracy in complex large-scale applications. Aiming at these challenges, an efficient enhanced chameleon swarm algorithm termed MCSA, combined with fractional-order calculus, sinusoidal adjustment of parameters and crossover-based comprehensive learning (CCL) strategy, is developed in this paper. Firstly, a good fractional-order calculus strategy is added to update the chameleon's attack velocity, which heightens the local search ability of CSA and accelerates the convergence speed of the algorithm; meanwhile, the sinusoidal adjustment of parameters is adopted to provide a better balance between exploration and exploitation of CSA. Secondly, the CCL strategy is used for the mutation to increase the diversity of the population and avoid becoming trapped in local optima. Three strategies enhance the overall performance and efficiency of the native CSA. Finally, the superiority of the presented MCSA is verified in detail by comparing it with native CSA and several state-of-the-art algorithms on the well-known 23 benchmark test functions, CEC2017 and CEC2019 test suites, respectively. Furthermore, the practicability of MCSA is also highlighted by six real-world engineering designs and two truss topology optimization problems. Simulation results demonstrate that MCSA has strong competitive capabilities and promising prospects. MCSA is potentially an excellent meta-heuristic algorithm for solving engineering optimization problems. [ABSTRACT FROM AUTHOR]

Due to the global progress, the optimization problems are becoming more and more complex. Hence, deterministic and heuristic approaches are no longer adequate for dealing with such sophisticated problems. subsequently, metaheuristics have recently emerged as an effective alternative for addressing the optimization problems. This paper proposes a novel metaheuristic called Young's Double-Slit Experiment (YDSE) optimizer, derived from a physical backdrop. The YDSE optimizer is inspired by Young's double-slit experiment, which is regarded as one of the most well-known classical physics experiments, revealing the wave nature of light. In YDSE optimizer, each fringe represents a possible solution in the population. Many concepts are modeled from the experiment, such as monochromatic light waves, Huygens' principle, constructive and destructive interference, wave intensity, amplitude, and path difference. The YDSE optimizer strikes a balance between exploration and exploitation by selecting either a constructive interference or a destructive interference based on the order number of the fringe. During the optimization process, the solution moves in search space based on its order number. If the solution has an odd number, it moves in the dark regions towards the central bright region, which is expected to contain the optimal solution. The algorithm exploits the promising areas in the bright fringe areas, which are assumed to contain the optimum. The performance of the YDSE optimizer is compared with another twelve metaheuristics using CEC 2014, CEC 2017, and CEC 2022. The benchmarks cover different unimodal, multimodal, hybrid, and composite test functions. Also, we consider ten constrained and unconstrained engineering optimization design problems. YDSE proved its superiority over the CEC 2014 and CEC 2017 winners, such as L-SHADE, LSHADE-cnEpSin, and LSHADE-SPACMA. The results and the statistical analysis demonstrated the outperformance of the proposed YDSE optimizer at a 95% confidence interval. • YDSE optimizer is a novel metaheuristic inspired by Young's double-slit experiment. • We compare YDSE to existing algorithms using CEC 2014, 2017, 2022 benchmarks. • Ten optimization problems are considered from the engineering field. • The statistical analysis demonstrated the superiority of the proposed YDSE. • YDSE optimizer outperforms L-SHADE, LSHADE-cnEpSin, and LSHADE-SPACMA. [ABSTRACT FROM AUTHOR]

The phase field method is proven to be a powerful tool in computational fracture mechanics in that it can predict crack nucleation and branching, in some cases, without explicitly tracking the crack path. However, in such applications as hydraulic fracturing or leakage, it is necessary or beneficial to have the crack geometry available. This paper presents a variational method to identify the crack path from phase field approaches to fracture. The method is proven to be successful not only for a simple curved crack but also for multiple and branched cracks. The algorithm employs the non-maximum suppression technique, a procedure borrowed from the image processing field, to detect a bounding area which covers the ridge of the phase field profile. After that, it is continued with the step to determine a cubic spline to represent the crack path and to improve it via a constrained optimization process. To demonstrate the performance of our method, we provide the results with three sets of representative examples. [ABSTRACT FROM AUTHOR]

A novel approach, which can control the minimum length scale in topology optimization in a straightforward and explicit way, is proposed in the present paper. This approach is constructed under the so-called Moving Morphable Components (MMC) based solution framework where optimized structural topology can be found by changing the shapes and layout of a set of trapezoid-shaped structural components on a fixed finite element mesh. The eXtended Finite Element Method (XFEM) is employed for structural response analysis and shape sensitivity information is obtained by carrying out numerical integration along the structural boundary. A precision definition of the minimum length scale of a structure is also suggested based on the proposed solution framework. Compared with existing approaches, the proposed approach can achieve minimum length scale control by simply setting lower bounds on a set of geometric design variables. Only a small number of purely geometric constraints imposed on the sizes of intersection regions should be considered explicitly in order to get a complete minimum length scale control. Numerical examples demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]