Owing to their excellent performance and large design space, curvilinear fiber-reinforced composite structures have gained considerable attention in engineering fields such as aerospace and automobile. In addition to the stiffness and strength of such structures, their stability also needs to be taken into account in the design. This study proposes a level-set-based optimization framework for maximizing the buckling load of curvilinear fiber-reinforced composite structures. In the proposed method, the contours of the level set function are used to represent fiber paths. For a composite laminate with a certain number of layers, one level set function is defined by radial basis functions and expansion coefficients for each layer. Furthermore, the fiber angle at an arbitrary point is the tangent orientation of the contour through this point. In the finite element of buckling, the stiffness and geometry matrices of an element are related to the fiber angle at the element centroid. This study considers the parallelism constraint for fiber paths. With the sensitivity calculation of the objective and constraint functions, the method of moving asymptotes is utilized to iteratively update all the expansion coefficients regarded as design variables. Two numerical examples under different boundary conditions are given to validate the proposed approach. Results show that the optimized curved fiber paths tend to be parallel and equidistant regardless of whether the composite laminates contain holes or not. Meanwhile, the buckling resistance of the final design is significantly improved.
This study proposes a B-spline-based multiresolution and multimaterial topology optimization (TO) design method for fail-safe structures (FSSs), aiming to achieve efficient and lightweight structural design while ensuring safety and facilitating the postprocessing of topological structures. The approach involves constructing a multimaterial interpolation model based on an ordered solid isotropic material with penalization (ordered-SIMP) that incorporates fail-safe considerations. To reduce the computational burden of finite element analysis, we adopt a much coarser analysis mesh and finer density mesh to discretize the design domain, in which the density field is described by the B-spline function. The B-spline can efficiently and accurately convert optimized FSSs into computer-aided design models. The 2D and 3D numerical examples demonstrate the significantly enhanced computational efficiency of the proposed method compared with the traditional SIMP approach, and the multimaterial TO provides a superior structural design scheme for FSSs. Furthermore, the postprocessing procedures are significantly streamlined.
Accurately controlling the nodal lines of vibrating structures with topology optimization is a highly challenging task. The major difficulties in this type of problem include a large number of design variables, the highly nonlinear and multi-peak characteristics of iteration, and the changeable orders of eigenmodes. In this study, an effective material-field series-expansion (MFSE)-based topology optimization design strategy for precisely controlling nodal lines is proposed. Here, two typical optimization targets are established: (1) minimizing the difference between structural nodal lines and their desired positions, and (2) keeping the position of nodal lines within the specified range while optimizing certain dynamic performance. To solve this complex optimization problem, the structural topology of structures is first represented by a few design variables on the basis of the MFSE model. Then, the problems are effectively solved using a sequence Kriging-based optimization algorithm without requiring design sensitivity analysis. The proposed design strategy inherently circumvents various numerical difficulties and can effectively obtain the desired vibration modes and nodal lines. Numerical examples are provided to validate the proposed topology optimization models and the corresponding solution strategy.
In this work, we put forward a massively efficient filter for topology optimization (TO) utilizing the splitting of tensor product structure. With the aid of splitting technique, the traditional weight matrices of B-splines and non-uniform rational B-spline implicit filters are decomposed equivalently into two or three submatrices, by which the sensitivity analysis is reformulated for the nodal design variables without altering the optimization process. Afterwards, an explicit sensitivity filter, which is decomposed by the splitting pipeline as that applied to implicit filter, is established in terms of the tensor product of the axial distances between adjacent element centroids, and the corresponding sensitivity analysis is derived for elemental design variables. According to the numerical results, the average updating time for the design variables is accelerated by two-order-of-magnitude for the decomposed filter compared with the traditional filter. In addition, the memory burden and computing time of the weight matrix are decreased by six- and three-order-of-magnitude for the decomposed filter. Therefore, the proposed filter is massively improved by the splitting of tensor product structure and delivers a much more efficient way of solving TO problems in the frameworks of isogeometric analysis and finite element analysis.
Piezoelectric actuators have received substantial attention among the industry and academia due to quick responses, such as high output force, high stiffness, high accuracy, and precision. However, the design of piezoelectric actuators always suffers from the emergence of several localized hinges with only one-node connection, which have difficulty satisfying manufacturing and machining requirements (from the over- or under-etching devices). The main purpose of the current paper is to propose a robust isogeometric topology optimization (RITO) method for the design of piezoelectric actuators, which can effectively remove the critical issue induced by one-node connected hinges and simultaneously maintain uniform manufacturability in the optimized topologies. In RITO, the isogeometric analysis replacing the conventional finite element method is applied to compute the unknown electro elastic fields in piezoelectric materials, which can improve numerical accuracy and then enhance iterative stability. The erode–dilate operator is introduced in topology representation to construct the eroded, intermediate, and dilated density distribution functions by non-uniform rational B-splines. Finally, the RITO formulation for the design of piezoelectric materials is developed, and several numerical examples are performed to test the effectiveness and efficiency of the proposed RITO method.
The variable density topology optimization (TO) method has been applied to various engineering fields because it can effectively and efficiently generate the conceptual design for engineering structures. However, it suffers from the problem of low continuity resulting from the discreteness of both design variables and explicit Heaviside filter. In this paper, an implicit Heaviside filter with high continuity is introduced to generate black and white designs for TO where the design space is parameterized by suitably graded truncated hierarchical B-splines (THB). In this approach, the fixed analysis mesh of isogeometric analysis is decoupled from the design mesh, whose adaptivity is implemented by truncated hierarchical B-spline subjected to an admissible requirement. Through the intrinsic local support and high continuity of THB basis, an implicit adaptively adjusted Heaviside filter is obtained to remove the checkboard patterns and generate black and white designs. Threefold advantages are attained in the proposed filter: a) The connection between analysis mesh and adaptive design mesh is easily established compared with the traditional adaptive TO method using nodal density; b) the efficiency in updating design variables is remarkably improved than the traditional implicit sensitivity filter based on B-splines under successive global refinement; and c) the generated black and white designs are preliminarily compatible with current commercial computer aided design system. Several numerical examples are used to verify the effectiveness of the proposed implicit Heaviside filter in compliance and compliant mechanism as well as heat conduction TO problems.
This paper proposes a novel method for the continuum topology optimization of transient vibration problem with maximum dynamic response constraint. An aggregated index in the form of an integral function is presented to cope with the maximum response constraint in the time domain. The density filter solid isotropic material with penalization method combined with threshold projection is developed. The sensitivities of the proposed index with respect to design variables are conducted. To reduce computational cost, the second-order Arnoldi reduction (SOAR) scheme is employed in transient analysis. Influences of aggregate parameter, duration of loading period, interval time, and number of basis vectors in the SOAR scheme on the final designs are discussed through typical examples while unambiguous configuration can be achieved. Through comparison with the corresponding static response from the final designs, the optimized results clearly demonstrate that the transient effects cannot be ignored in structural topology optimization.
This paper presents a MATLAB implementation of the material-field series-expansion (MFSE) topo-logy optimization method. The MFSE method uses a bounded material field with specified spatial correlation to represent the structural topology. With the series-expansion method for bounded fields, this material field is described with the characteristic base functions and the corresponding coefficients. Compared with the conventional density-based method, the MFSE method decouples the topological description and the finite element discretization, and greatly reduces the number of design variables after dimensionality reduction. Other features of this method include inherent control on structural topological complexity, crisp structural boundary description, mesh independence, and being free from the checkerboard pattern. With the focus on the implementation of the MFSE method, the present MATLAB code uses the maximum stiffness optimization problems solved with a gradient-based optimizer as examples. The MATLAB code consists of three parts, namely, the main program and two subroutines (one for aggregating the optimization constraints and the other about the method of moving asymptotes optimizer). The implementation of the code and its extensions to topology optimization problems with multiple load cases and passive elements are discussed in detail. The code is intended for researchers who are interested in this method and want to get started with it quickly. It can also be used as a basis for handling complex engineering optimization problems by combining the MFSE topology optimization method with non-gradient optimization algorithms without sensitivity information because only a few design variables are required to describe relatively complex structural topology and smooth structural boundaries using the MFSE method.
In structural design optimization involving transient responses, time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis. In this work, the influence of time integration scheme is studied numerically for the adjoint shape sensitivity analysis of two benchmark transient heat conduction problems within the framework of isogeometric analysis. It is found that (i) the explicit approach (
Advanced manufacturing processes such as additive manufacturing offer now the capability to control material placement at unprecedented length scales and thereby dramatically open up the design space. This includes the considerations of new component topologies as well as the architecture of material within a topology offering new paths to creating lighter and more efficient structures. Topology optimization is an ideal tool for navigating this multiscale design problem and leveraging the capabilities of advanced manufacturing technologies. However, the resulting design problem is computationally challenging as very fine discretizations are needed to capture all micro-structural details. In this paper, a method based on reduction techniques is proposed to perform efficiently topology optimization at multiple scales. This method solves the design problem without length scale separation, i.e., without iterating between the two scales. Ergo, connectivity between space-varying micro-structures is naturally ensured. Several design problems for various types of micro-structural periodicity are performed to illustrate the method, including applications to infill patterns in additive manufacturing.
We present an energy penalization method for isogeometric topology optimization using moving morphable components (ITO–MMC), propose an ITO–MMC with an additional bilateral or periodic symmetric constraint for symmetric structures, and then extend the proposed energy penalization method to an ITO–MMC with a symmetric constraint. The energy penalization method can solve the problems of numerical instability and convergence for the ITO–MMC and the ITO–MMC subjected to the structural symmetric constraint with asymmetric loads. Topology optimization problems of asymmetric, bilateral symmetric, and periodic symmetric structures are discussed to validate the effectiveness of the proposed energy penalization approach. Compared with the conventional ITO–MMC, the energy penalization method for the ITO–MMC can improve the convergence rate from 18.6% to 44.5% for the optimization of the asymmetric structure. For the ITO–MMC under a bilateral symmetric constraint, the proposed method can reduce the objective value by 5.6% and obtain a final optimized topology that has a clear boundary with decreased iterations. For the ITO–MMC under a periodic symmetric constraint, the proposed energy penalization method can dramatically reduce the number of iterations and obtain a speedup of more than 2.
This paper presents a new robust topology optimization framework for hinge-free compliant mecha- nisms with spatially varying material uncertainties, which are described using a non-probabilistic bounded field model. Bounded field uncertainties are efficiently represented by a reduced set of uncertain-but-bounded coefficients on the basis of the series expansion method. Robust topology optimization of compliant mechanisms is then defined to minimize the variation in output displacement under constraints of the mean displacement and predefined material volume. The nest optimization problem is solved using a gradient-based optimization algorithm. Numerical examples are presented to illustrate the effectiveness of the proposed method for circumventing hinges in topology optimization of compliant mechanisms.
In recent years, the new technologies and discoveries on manufacturing materials have encouraged researchers to investigate the appearance of material properties that are not naturally available. Materials featuring a specific stiffness, or structures that combine non-structural and structural functions are applied in the aerospace, electronics and medical industry fields. Particularly, structures designed for dynamic actuation with reduced vibration response are the focus of this work. The bi-material and multifunctional concepts are considered for the design of a controlled piezoelectric actuator with vibration suppression by means of the topology optimization method (TOM). The bi-material piezoelectric actuator (BPEA) has its metallic host layer designed by the TOM, which defines the structural function, and the electric function is given by two piezo-ceramic layers that act as a sensor and an actuator coupled with a constant gain active velocity feedback control (AVFC). The AVFC, provided by the piezoelectric layers, affects the structural damping of the system through the velocity state variables readings in time domain. The dynamic equation analyzed throughout the optimization procedure is fully elaborated and implemented. The dynamic response for the rectangular four-noded finite element analysis is obtained by the Newmark’s time-integration method, which is applied to the physical and the adjoint systems, given that the adjoint formulation is needed for the sensitivity analysis. A gradient-based optimization method is applied to minimize the displacement energy output measured at a predefined degree-of-freedom of the BPEA when a transient mechanical load is applied. Results are obtained for different control gain values to evaluate their influence on the final topology.
Regularization of the level-set (LS) field is a critical part of LS-based topology optimization (TO) approaches. Traditionally this is achieved by advancing the LS field through the solution of a Hamilton-Jacobi equation combined with a reinitialization scheme. This approach, however, may limit the maximum step size and introduces discontinuities in the design process. Alternatively, energy functionals and intermediate LS value penalizations have been proposed. This paper introduces a novel LS regularization approach based on a signed distance field (SDF) which is applicable to explicit LS-based TO. The SDF is obtained using the heat method (HM) and is reconstructed for every design in the optimization process. The governing equations of the HM, as well as the ones describing the physical response of the system of interest, are discretized by the extended finite element method (XFEM). Numerical examples for problems modeled by linear elasticity, nonlinear hyperelasticity and the incompressible Navier-Stokes equations in two and three dimensions are presented to show the applicability of the proposed scheme to a broad range of design optimization problems.
Enabled by advancements in multi-material additive manufacturing, lightweight lattice structures consisting of networks of periodic unit cells have gained popularity due to their extraordinary performance and wide array of functions. This work proposes a density-based robust topology optimization method for meso- or macro-scale multi-material lattice structures under any combination of material and load uncertainties. The method utilizes a new generalized material interpolation scheme for an arbitrary number of materials, and employs univariate dimension reduction and Gauss-type quadrature to quantify and propagate uncertainty. By formulating the objective function as a weighted sum of the mean and standard deviation of compliance, the tradeoff between optimality and robustness can be studied and controlled. Examples of a cantilever beam lattice structure under various material and load uncertainty cases exhibit the efficiency and flexibility of the approach. The accuracy of univariate dimension reduction is validated by comparing the results to the Monte Carlo approach.
In this paper, a parametric level-set-based topology optimization framework is proposed to concurrently optimize the structural topology at the macroscale and the effective infill properties at the micro/meso scale. The concurrent optimization is achieved by a computational framework combining a new parametric level set approach with mathematical programming. Within the proposed framework, both the structural boundary evolution and the effective infill property optimization can be driven by mathematical programming, which is more advantageous compared with the conventional partial differential equation-driven level set approach. Moreover, the proposed approach will be more efficient in handling nonlinear problems with multiple constraints. Instead of using radial basis functions (RBF), in this paper, we propose to construct a new type of cardinal basis functions (CBF) for the level set function parameterization. The proposed CBF parameterization ensures an explicit impose of the lower and upper bounds of the design variables. This overcomes the intrinsic disadvantage of the conventional RBF-based parametric level set method, where the lower and upper bounds of the design variables oftentimes have to be set by trial and error. A variational distance regularization method is utilized in this research to regularize the level set function to be a desired distance-regularized shape. With the distance information embedded in the level set model, the wrapping boundary layer and the interior infill region can be naturally defined. The isotropic infill achieved via the mesoscale topology optimization is conformally fit into the wrapping boundary layer using the shape-preserving conformal mapping method, which leads to a hierarchical physical structure with optimized overall topology and effective infill properties. The proposed method is expected to provide a timely solution to the increasing demand for multiscale and multifunctional structure design.
Maximizing the fundamental eigenfrequency is an efficient means for vibrating structures to avoid resonance and noises. In this study, we develop an isogeometric analysis (IGA)-based level set model for the formulation and solution of topology optimization in cases with maximum eigenfrequency. The proposed method is based on a combination of level set method and IGA technique, which uses the non-uniform rational B-spline (NURBS), description of geometry, to perform analysis. The same NURBS is used for geometry representation, but also for IGA-based dynamic analysis and parameterization of the level set surface, that is, the level set function. The method is applied to topology optimization problems of maximizing the fundamental eigenfrequency for a given amount of material. A modal track method, that monitors a single target eigenmode is employed to prevent the exchange of eigenmode order number in eigenfrequency optimization. The validity and efficiency of the proposed method are illustrated by benchmark examples.
This paper presents a manufacturing cost constrained topology optimization algorithm considering the laser powder bed additive manufacturing process. Topology optimization for additive manufacturing was recently extensively studied, and many related topics have been addressed. However, metal additive manufacturing is an expensive process, and the high manufacturing cost severely hinders the widespread use of this technology. Therefore, the proposed algorithm in this research would provide an opportunity to balance the manufacturing cost while pursuing the superior structural performance through topology optimization. Technically, the additive manufacturing cost model for laser powder bed-based process is established in this paper and real data is collected to support this model. Then, this cost model is transformed into a level set function-based expression, which is integrated into the level set topology optimization problem as a constraint. Therefore, by properly developing the sensitivity result, the metallic additive manufacturing part can be optimized with strictly constrained manufacturing cost. Effectiveness of the proposed algorithm is proved by numerical design examples.
The advances of manufacturing techniques, such as additive manufacturing, have provided unprecedented opportunities for producing multiscale structures with intricate latticed/cellular material microstructures to meet the increasing demands for parts with customized functionalities. However, there are still difficulties for the state-of-the-art multiscale topology optimization (TO) methods to achieve manufacturable multiscale designs with cellular materials, partially due to the disconnectivity issue when tiling material microstructures. This paper attempts to address the disconnectivity issue by extending component-based TO methodology to multiscale structural design. An effective linkage scheme to guarantee smooth transitions between neighboring material microstructures (unit cells) is devised and investigated. Associated with the advantages of components-based TO, the number of design variables is greatly reduced in multiscale TO design. Homogenization is employed to calculate the effective material properties of the porous materials and to correlate the macro/structural scale with the micro/material scale. Sensitivities of the objective function with respect to the geometrical parameters of each component in each material microstructure have been derived using the adjoint method. Numerical examples demonstrate that multiscale structures with well-connected material microstructures or graded/layered material microstructures are realized.
The level set method (LSM), which is transplanted from the computer graphics field, has been successfully introduced into the structural topology optimization field for about two decades, but it still has not been widely applied to practical engineering problems as density-based methods do. One of the reasons is that it acts as a boundary evolution algorithm, which is not as flexible as density-based methods at controlling topology changes. In this study, a level set band method is proposed to overcome this drawback in handling topology changes in the level set framework. This scheme is proposed to improve the continuity of objective and constraint functions by incorporating one parameter, namely, level set band, to seamlessly combine LSM and density-based method to utilize their advantages. The proposed method demonstrates a flexible topology change by applying a certain size of the level set band and can converge to a clear boundary representation methodology. The method is easy to implement for improving existing LSMs and does not require the introduction of penalization or filtering factors that are prone to numerical issues. Several 2D and 3D numerical examples of compliance minimization problems are studied to illustrate the effects of the proposed method.
Topology optimization is a pioneer design method that can provide various candidates with high mechanical properties. However, high resolution is desired for optimum structures, but it normally leads to a computationally intractable puzzle, especially for the solid isotropic material with penalization (SIMP) method. In this study, an efficient, high-resolution topology optimization method is developed based on the super-resolution convolutional neural network (SRCNN) technique in the framework of SIMP. SRCNN involves four processes, namely, refinement, path extraction and representation, nonlinear mapping, and image reconstruction. High computational efficiency is achieved with a pooling strategy that can balance the number of finite element analyses and the output mesh in the optimization process. A combined treatment method that uses 2D SRCNN is built as another speed-up strategy to reduce the high computational cost and memory requirements for 3D topology optimization problems. Typical examples show that the high-resolution topology optimization method using SRCNN demonstrates excellent applicability and high efficiency when used for 2D and 3D problems with arbitrary boundary conditions, any design domain shape, and varied load.