Explicit Topology Optimization Research Articles

Topology optimization for the design of manufacturable piezoelectric energy harvesters using dual-MMC method

Abstract This paper presents a new topology optimization scheme for the manufacturable piezoelectric energy harvesters (PEHs). Most of the existing topology optimization schemes for the design for PEHs are difficult to cope with manufacturing constraints producing design results that pose serious challenges for the local poling of the piezoelectric materials. In this work, dual-Moving Morphable Component (dual-MMC) scheme for explicit topology optimization for the design of PEHs is presented. In dual-MMC scheme, two independent sets of MMC are employed to describe the structural topology of the PEH and polarization profile in piezoelectric material in an explicit manner. With the use of the scheme, the shape of electrodes and the opposite polarization directions in the local poling process can be effectively treated as a constraint making the realization of the PEH an easy task. Several examples for the design of cantilever-type PEH are provided to demonstrate the effectiveness of the proposed approach. Furthermore, a designed PEH actually manufactured for demonstration of the production process.

Moving morphable component (MMC) topology optimization with different void structure scaling factors.

The explicit topology optimization method based on moving morphable component (MMC) has attracted more and more attention, and components are the basic building blocks of the implementation of MMC method. In the present work, a MMC topology optimization method based on component with void structure is followed with interest. On the basis of analyzing the characteristics of components used by MMC method, the topology description function for component with void structure is presented, where a quantitative scaling factor is introduced without increasing the number of design variables. Taking the minimum flexibility as the optimization objective, an example of short beam is analyzed with different void structure scaling factors. The results show that different scaling factors have a greater impact on the final topology optimization structure, and an ideal topology structure can be obtained with an appropriate scaling factor. Finally, some problems in the optimization process are analyzed and indicate that appropriate mesh density should be chose for component with void structure in order to achieve good optimization results.

Explicit Topology Optimization of Voronoi Foams.

Topology optimization can maximally leverage the high DOFs and mechanical potentiality of porous foams but faces challenges in adapting to free-form outer shapes, maintaining full connectivity between adjacent foam cells, and achieving high simulation accuracy. Utilizing the concept of Voronoi tessellation may help overcome the challenges owing to its distinguished properties on highly flexible topology, natural edge connectivity, and easy shape conforming. However, a variational optimization of the so-called Voronoi foams has not yet been fully explored. In addressing the issue, a concept of explicit topology optimization of open-cell Voronoi foams is proposed that can efficiently and reliably guide the foam's topology and geometry variations under critical physical and geometric requirements. Taking the site (or seed) positions and beam radii as the DOFs, we explore the differentiability of the open-cell Voronoi foams w.r.t. its seed locations, and propose a highly efficient local finite difference method to estimate the derivatives. During the gradient-based optimization, the foam topology can change freely, and some seeds may even be pushed out of shape, which greatly alleviates the challenges of prescribing a fixed underlying grid. The foam's mechanical property is also computed with a much-improved efficiency by an order of magnitude, in comparison with benchmark FEM, via a new material-aware numerical coarsening method on its highly heterogeneous density field counterpart. We show the improved performance of our Voronoi foam in comparison with classical topology optimization approaches and demonstrate its advantages in various settings.

Graded infill lattice structures design based on the moving morphable component method and partitioned coordinate mapping technique

This work presents a novel method for designing gradient-varying lattice structures based on explicit topology optimization. The proposed method employs a set of Moving Morphable Components (MMC) and NURBS-based partitioned coordinate mapping (PCM) functions to describe material distribution within the design domain and control points grid in the parametric space for local control of the lattice structure within each subdomain, thereby generating diverse microstructure topologies in different regions while being able to ensure clear structural boundaries and high-order smooth connections between adjacent microstructures. Since the optimization parameters include only the explicit geometric parameters of the components and the perturbation coefficients of the mapping function, the number of design variables is substantially smaller compared to existing algorithms. The proposed method can efficiently generate graded lattice structures for various loading conditions and graded-infill structures. Numerical results validate the effectiveness of the proposed method.

Explicit topology optimization of three-dimensional geometrically nonlinear structures

Explicit topology optimization of three-dimensional geometrically nonlinear structures

Higher-order topological insulators by ML-enhanced topology optimization

Energy in lower dimensions, such as edges or corners, can be robustly confined by the higher-order topological insulators (HOTIs) against defects. Being promising for wave manipulation, the unique property of HOTIs is protected by the associated topological invariant and bulk band. Nevertheless, it is still challenging to fast design topologically nontrivial continuum unit cells with highly localized edge/corner modes in photonic and phononic systems. In the present work, with the help of a multitask learning model simultaneously predicting the discrete-valued topological invariant and continuum-valued bandgap range, a fast design framework is developed using an explicit topology optimization method for mechanical HOTIs. In the solution process, the machine learning model and backpropagation algorithm are integrated into a MultiStart solver to accelerate the design efficiency by 3–4 orders than the traditional topology optimization method. Furthermore, a novel programmable mechanical imaging device illustrates the applications of the optimized HOTI with highly localized corner states. This AI-enhanced design paradigm can be easily extended for the fast design of optimized topological materials among various physical systems.

Deep learning driven real time topology optimization based on improved convolutional block attention (Cba-U-Net) model

Topology optimization design provides innovative structures with excellent thermal, mechanical and acoustic performance for modern engineering. Moving Morphable Component (MMC), as an emerging explicit topology optimization method, can effectively avoid many optimization problems such as the checkerboard phenomenon, however, its optimization iteration process still consumes considerable time, which makes real-time structural topology optimization impossible. Therefore, a lightweight and high-efficiency convolutional neural network, the improved convolutional block attention U-Net (Cba-U-Net) model, is proposed for topology-optimized configuration prediction, which avoids its own tedious iterative computation process and acquires the topology configuration in real-time. It is demonstrated that the proposed network not only obtains accurate topology-optimized configurations in negligible time but also has an accuracy rate of 91.42% compared to other deep learning models. The improved Cba-U-Net model is suitable not only for Moving Morphable Components but also for other optimization algorithms, such as Solid Isotropic Material with Penalization (SIMP) and Evolutionary Structural Optimization Method (ESO). By combining deep learning with topological optimization algorithms, this form of optimization is highly generalizable for practical large-scale projects.

A study on topology optimization of heat dissipation structures with different objective functions based on an explicit moving morphable components method

This article investigates two different objective functions (minimizing heat dissipation weakness and temperature variance) to realize topology design optimization of heat dissipation problems based on the explicit topology optimization method of moving morphable components (MMC). The advantages of explicit and clear geometric profile descriptions in dealing with heat dissipation are explored. Sensitivity derivation with respect to the explicit topological geometric variables of the MMC-based method is established for two objective functions. Numerical examples are presented, with extensive discussion and comparisons of the key factors of maximum temperature gradient, temperature variance and maximum temperature based on the two objective functions. These discussions demonstrate the scope of application of the objective functions. Compared with density-based topology design optimization, the present MMC-based method describes the geometric profile of the heat dissipation structure accurately and achieves a more accurate and efficient heat dissipation performance of the structure.

An efficient and easy-to-extend Matlab code of the Moving Morphable Component (MMC) method for three-dimensional topology optimization

Explicit topology optimization methods have received ever-increasing interest in recent years. In particular, a 188-line Matlab code of the two-dimensional (2D) Moving Morphable Component (MMC)-based topology optimization method was released by Zhang et al. (Struct Multidiscip Optim 53(6):1243-1260, 2016). The present work aims to propose an efficient and easy-to-extend 256-line Matlab code of the MMC method for three-dimensional (3D) topology optimization implementing some new numerical techniques. To be specific, by virtue of the function aggregation technique, accurate sensitivity analysis, which is also easy-to-extend to other problems, is achieved. Besides, based on an efficient identification algorithm for load transmission path, the degrees of freedoms (DOFs) not belonging to the load transmission path are removed in finite element analysis (FEA), which significantly accelerates the optimization process. As a result, compared to the corresponding 188-line 2D code, the performance of the optimization results, the computational efficiency of FEA, and the convergence rate and the robustness of optimization process are greatly improved. For the sake of completeness, a refined 218-line Matlab code implementing the 2D-MMC method is also provided.

Flexoelectric nanostructure design using explicit topology optimization

Flexoelectricity is the coupling between polarization and strain gradient. As the large strain gradient leads to a strong flexoelectric effect, the design of flexoelectric nano-structure via topology optimization has seen growing attentions. In the present work, an explicit topology optimization framework is proposed for flexoelectric structures design. To achieve this purpose, the Moving Morphable Void (MMV)-based approach is employed in the context of Isogeometric Analysis (IGA) combined with the Trimming Surface Analysis (TSA). Energy conversion factor and effective electric polarizability are respectively optimized to improve the flexoelectric performance of the nanostructure. Performing design under the explicit framework coupled with IGA can bring several advantages. Due to the use of NURBS basis functions of IGA, the required continuity in the approximation of the PDEs of flexoelectricity can be satisfied straightforwardly. Furthermore, with the use of the TSA technique, the occurrence of weak/gray material, which may cause numerical instability in flexoelectricity design, can be avoided. Also due to the explicit geometry description in MMV, the optimized result can be imported to the CAD system directly, which is significant for nanoscale structure from manufacturing perspective. Several representative numerical examples for topology optimization of flexoelectric structures are presented to demonstrate the effectiveness and advantages of the proposed approach.

Design of compliant mechanisms: An explicit topology optimization method using end-constrained spline curves with variable width

In this study, an approach using moving end-constrained spline (ECS) curves with variable widths is presented for the explicit topology optimization of compliant mechanisms. The proposed approach uses a series of spline curves to represent the topology of the compliant mechanism. Instead of using a fixed width, we let the width of such spline curves vary along the curve. The optimized design is obtained by optimizing the width of the spline curves and the position of the control points. To avoid infeasible solutions in the design process, the end points of several specific spline curves are constrained according to the design region. The validity of the proposed method is demonstrated using different types of splines, such as the Bézier curve and B-spline curve, for the design of a force inverter, a compliant gripper and a compliant orthogonal displacement amplification mechanism.

Topology optimization for disc structures with buckling and displacement constraints

This article presents a novel topology optimization framework, which takes the key nodal displacements and the critical buckling load factor as constraints to guarantee that the designs of torsional disc structures not only satisfy the resistant capability of in-plane deformation but also do not cause out-of-plane buckling failure. The composite exponential filter functions and Taylor expansion are applied to obtain the approximate explicit functions between structural performance and topological design variables based on the results of static and buckling analyses. The dual-sequence quadratic programming algorithm is applied to solve the explicit topology optimization model. In addition, an adaptive image filter technology is developed to overcome the numerical instability problem that occurs in the finite element model of discs with significant changes in mesh size. Finally, some designs of torsional disc structures are presented to illustrate the effectiveness and feasibility of this method.

An Approach for Geometrically Nonlinear Topology Optimization Using Moving Wide-Bézier Components With Constrained Ends

Abstract Standard moving morphable component (MMC)-based topology optimization methods use free components with explicitly geometrical parameters as design units to obtain the optimal structural topology by moving, deforming, and covering such components. In this study, we intend to present a method for geometrically nonlinear explicit topology optimization using moving wide-Bézier components with constrained ends. Not only can the method efficiently avoid the convergence issues associated with nonlinear structural response analysis, but it can also alleviate the component disconnection issues associated with the standard MMC-based topology optimization methods. The numerical investigations proposed in this work indicate that the proposed method allows us to obtain results in accordance with the current literature with a more stable optimization process. In addition, the proposed method can easily achieve minimum length scale control without adding constraints.

Explicit structural topology optimization using boundary element method‐based moving morphable void approach

AbstractThis article presents an explicit topology optimization approach using boundary element method based moving morphable void (MMV). Structural analysis in the conventional MMV approach is performed on a fixed mesh over the entire design domain. However, in the proposed method structural boundaries are described using B‐splines and discretized with boundary elements making explicit representation of the boundaries possible. In this regard, the proposed approach has following merits: a) the use of weak material, which is often adopted to mimic voids can be completely avoided; b) exact and explicit description of voids in MMV allows the capture of tiny structural features in a robust and effective way; c) numerical instabilities stemming from the use of fixed meshes can be naturally avoided. Several numerical examples are provided to demonstrate the effectiveness of the proposed approach.

A component-based method for the optimization of stiffener layout on large cylindrical rib-stiffened shell structures

In the present work, an optimization method is proposed in order to produce innovative stiffening layouts for large stiffened cylindrical shell structures, as they appear in the aerospace industry. A component-based logic is applied on a ground mesh of structural elements (shells and beams), which was inspired by techniques of explicit topology optimization on solid elements (plane or tridimensional massive models). Geometric components, representing the layout of the stiffeners (i.e. location, shape and size), are projected onto the ground mesh, resulting in controlled sets of active beam elements. These sets constitute the structural representation of the stiffeners’ layout in the optimization model, which is then used to evaluate the objective and constraint functions of the optimization problem as well as their semi-analytical sensitivities. By applying the optimization method to compliance minimization problems, we show the efficiency and accuracy of the proposed method and its capability to handle a typical aerospace structure, such as a space-launcher part: a stiffened cylindrical shell in presence of an access hatch.

A scaled boundary finite element based explicit topology optimization approach for three‐dimensional structures

SummaryThis article proposes an efficient approach for solving three‐dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)‐based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.

Explicit topology optimization of nearly incompressible materials using polytopal composite elements

In this paper, an efficient explicit topology optimization approach to nearly incompressible materials was proposed. The crucial idea is based on a smooth projection of discrete moving morphable bars (MMBs) onto an analytical grid of polytopal composite finite elements (PCEs). The PCEs, which are built with piecewise linear shape functions and assumed strain fields, aim to mitigate the inherent volumetric locking issue and accurately model nearly incompressible materials. Unlike most existing techniques that use the mixed displacement-pressure formulation and conventional topology optimization, the current method is based on the pure displacement formulation and explicit topology optimization with only hundreds of design variables. Numerical results demonstrated that the proposed approach works stably and produces proper optimal designs of nearly incompressible structures without the mesh-dependence while filtering techniques or perimeter constraints are not required. The investigation promises the potential for solving a wide range of incompressible problems.

Explicit multi-material topology optimization embedded with variable-size movable holes using moving morphable bars

In this article, an explicit topology optimization method is proposed for maximizing the overall stiffness of a multi-material continuum structure embedded with multiple variable-size movable holes based on the Moving Morphable Bars (MMB) method. To this end, multiple sets of morphable bars are used to describe the structural topology of multiphase materials, and level-set functions are used to describe the geometrical shapes of variable-size movable holes. The morphable bars and embedded holes are then projected onto multiple density fields to avoid remeshing the grids. A Solid Isotropic Material with Penalization (SIMP)-like multi-material interpolation scheme that considers embedding holes is introduced for material parameterization. Several numerical examples are performed to illustrate the effectiveness of the proposed formulation. It is seen that far fewer design variables are involved in the present formulation for multi-material topology optimization when compared to the existing optimization framework with embedded movable holes.

Generation of strut-and-tie models in concrete structures by topology optimization based on moving morphable components

Explicit topology optimization based on moving morphable components (MMCs) is a novel and revolutionary topology optimization framework that may be exploited for the generation of strut-and-tie models (STMs) in concrete structures. The article deals with the generation of STMs by solving a compliance minimization problem subject to a volume constraint. The feasible STMs are generated by a set of components that can move, dilate, shrink, overlap and merge. A new extraction method is adopted to address the crucial problem that interprets the topology optimization results as the STM. Design examples of deep beams and corbels are presented to demonstrate the validity and effectiveness of the MMC-based topology optimization method as a rational and reliable design tool for concrete structures. Moreover, the high efficiency and the applicability of the proposed approach to the multiple load case are also investigated.

An explicit topology optimization method using moving polygonal morphable voids (MPMVs)

Introduction: Conventional topology optimization approaches are implemented in an implicit manner with a very large number of design variables, requiring large storage and computation costs. In this study, an explicit topology optimization approach is proposed by movonal morphable voids whose geometry parameters are considered as design variables.
 Methods: Each polygonal void plays as an empty-material zone that can move, change its shapes, and overlap with its neighbors in a design space. The geometry eters of MPMVs consisting of the coordinates of polygonal vertices are utilized to render the structure in the design domain in an element density field. The density function of the elements located inside polygonal voids is described by a smooth exponential function that allows utilizing gradient-based optimization solvers.
 Results & Conclusion: Compared with conventional topology optimization approaches, the MPMV approach uses fewer design variables, ensure mesh-independence solution without filtering techniques or perimeter constraints. Several numerical examples are solved to validate the efficiency of the MPMV approach.