Topology Optimization Framework Research Articles

Topology optimization considering the Drucker–Prager criterion with a surrogate nonlinear elastic constitutive model

We address material nonlinear topology optimization problems considering the Drucker–Prager strength criterion by means of a surrogate nonlinear elastic model. The nonlinear material model is based on a generalized J2 deformation theory of plasticity. From an algorithmic viewpoint, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. The objective function of the optimization problem consists of maximizing the strain energy of the system in equilibrium subjected to a volume constraint. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved through direct minimization of the structural strain energy using Newton’s method with an inexact line search strategy. Four numerical examples demonstrate features of the proposed nonlinear topology optimization framework considering the Drucker–Prager strength criterion.

Data-Driven Additive Manufacturing Constraints for Topology Optimization

Abstract The topology optimization (TO) of structures to be produced using additive manufacturing (AM) is explored using a data-driven constraint function that predicts the minimum producible size of small features in different shapes and orientations. This shape- and orientation-dependent manufacturing constraint, derived from experimental data, is implemented within a TO framework using a modified version of the moving morphable components (MMC) approach. Because the analytic constraint function is fully differentiable, gradient-based optimization can be used. The MMC approach is extended in this work to include a “bootstrapping” step, which provides initial component layouts to the MMC algorithm based on intermediate solid isotropic material with penalization (SIMP) topology optimization results. This “bootstrapping” approach improves convergence compared with reference MMC implementations. Results from two compliance design optimization example problems demonstrate the successful integration of the manufacturability constraint in the MMC approach, and the optimal designs produced show minor changes in topology and shape compared to designs produced using fixed-radius filters in the traditional SIMP approach. The use of this data-driven manufacturability constraint makes it possible to take better advantage of the achievable complexity in additive manufacturing processes, while resulting in typical penalties to the design objective function of around only 2% when compared with the unconstrained case.

Topology-Finding of Tensegrity Structures Considering Global Stability Condition

This study proposes a general framework for topology-finding or topology optimization of tensegrity structures. The existing topology-finding formulation of tensegrity structures based on mixed-integer linear programming (MILP) was improved and transformed into a formulation based on mixed-integer semidefinite programming (MISDP) which considers the global stability condition of tensegrity. We illustrated and analyzed two undesirable phenomena, loss of prestress stability and loss of integrity, caused by the missing global stability condition in previous MILP-based approaches. A branch-and-bound algorithm combined with a primal-dual interior-point algorithm is employed to solve the proposed MISDP model. Some numerical examples illustrated the improvements and effectiveness of the proposed approach. The proposed approach successfully can avoid the two undesirable phenomena and ensure the global stability of the found tensegrity structures. By using different stability conditions in the topology-finding process, the proposed approach can find general stable tensegrity structures and superstable tensegrity structures.

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.

Spatial-varying multi-phase infill design using density-based topology optimization

Porous infill structures have been widely studied and used in additive manufacturing because of their lightweight and excellent mechanical properties. However, without considering graded multi-material infill pattern, existed infill design methods have not yet fully tapped the potential of multiple materials in structural design. In this paper, a systematic multi-phase infill design method is proposed to generate graded multi-material infill structures. The method builds upon a unified multi-material density-based topology optimization framework, in which a modified multi-phase material interpolation formulation is presented to represent the relationship between the stiffness matrix and design variables. To generate spatial-varying and multi-phase structures distributed in the interior of a design domain, the maximum local material volume constraints are imposed on each phase material in the neighborhood of each element in the design domain. A series of relaxations is introduced into the framework to facilitate the implementation of the gradient-based optimization algorithm. The whole design process is performed on a full-size finite element analysis, so it can avoid the separation of scales and naturally guarantee the optimized infills to be smoothly connected. The applicability and effectiveness of the proposed multi-phase infill generation model are then demonstrated by several typical numerical examples with the objective of minimum compliance.

Topology optimization of periodic barriers for surface waves

Dispersion engineering is always the important topic in the field of artificial periodic structures. In particular, topology optimization of composite structures with expected bandgaps plays a key role. However, most reported studies focused on topology optimization for bulk waves, and the optimization for surface wave bandgaps (SWBGs) is still missing. In this paper, we develop a topology optimization framework based on the genetic algorithm and finite element method to design periodic barriers embedded in semi-infinite space for reducing surface waves on demand. The objective functions for SWBGs are proposed based on the energy distribution properties of surface waves. The numerical results show that the optimization framework has stable convergence for this problem and is effective to optimize SWBGs. Considering large SWBGs and low filling fraction of solids, we investigate single- and multi-objective optimizations, respectively, and obtain novel wave barriers with good performance. The beneficial configuration features and mechanism of broadband SWBGs from the optimized results are explored. The results indicate that for the first-order SWBG, most optimized structures consist of a main scatterer at the center and some subsidiary scatterers near the surface. The rigid body resonance of the main scatterer determines the lower edge of SWBG, and the subsidiary scatterers can regulate the upper edge. Higher-order SWBGs are generated from the interaction of multiple scatterers, whose relative distance has a great influence on the position of SWBGs. The optimized structures can make the surface waves propagate far away from the surface within the frequencies of bandgaps, leading to a strong attenuation of surface vibration. In practice, our topological optimization framework is promising in designing high-performance surface wave devices and novel isolating structures for earthquake or environmental vibration in civil engineering.

Topology Optimization of Energy-Dissipating Plastic Structures with Shear Modified Gurson–Tvergaard–Needleman Model

AbstractThis paper presents a density-based topology optimization framework for designing energy-dissipating plastic structures. In order to mitigate the material damage during the plastic energy d...

Sequentially coupled gradient-based topology and domain shape optimization

A coupled topology and domain shape optimization framework is presented that is based on incorporating the shape design variables of the design domain in the Solid Isotropic Material with Penalization topology optimization method. The shape and topology design variables are incrementally updated in a sequential fashion, using a staggered numerical update scheme. Non-Uniform Rational B-Splines are employed to parameterize the shape of the design domain. This not only guarantees a highly accurate description of the shape boundaries by means of smooth basis functions with compact support, but also enables an efficient control of the design domain with only a few control points. Furthermore, the optimization process is performed in a computationally efficient way by applying a gradient-based optimization algorithm, for which the sensitivities can be computed in closed form. The usefulness of the coupled optimization approach is demonstrated by analyzing several benchmark problems that are subjected to different types of initial conditions and domain bounds. The variation in simulation results denotes that a careful construction of the initial design domain is necessary and meaningful.

Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach

We introduce a general multi-material topology optimization framework for large deformation problems that effectively handles an arbitrary number of candidate hyperelastic materials and addresses three major associated challenges: material interpolation, excessive distortion of low-density elements, and computational efficiency. To account for many nonlinear elastic materials, we propose a material interpolation scheme that, instead of interpolating multiple material parameters (such as Young’s modulus), interpolates multiple nonlinear stored-energy functions. To circumvent convergence difficulties caused by excessive distortions of low-density elements under large deformations, an energy interpolation scheme is revisited to account for multiple candidate hyperelastic materials. Computational efficiency is addressed from both structural analysis and optimization perspectives. To solve the nonlinear state equations efficiently, we employ the lower-order Virtual Element Method in conjunction with tailored adaptive mesh refinement and coarsening strategies. To efficiently update the design variables of the multi-material system, we exploit the separable nature and improve the ZPR (Zhang–Paulino–Ramos) update scheme to account for positive sensitivities and update the design variables associated with each volume constraint in parallel. Four design examples with three types of nonlinear material models demonstrate the efficiency and effectiveness of the proposed framework.

Anisotropic Multicomponent Topology Optimization for Additive Manufacturing With Build Orientation Design and Stress-Constrained Interfaces

Abstract This paper presents a multicomponent topology optimization method for designing structures assembled from additively manufactured components, considering anisotropic material behavior for each component due to its build orientation, distinct material behavior, and stress constraints at component interfaces (i.e., joints). Based upon the multicomponent topology optimization (MTO) framework, the simultaneous optimization of structural topology, its partitioning, and the build orientations of each component is achieved, which maximizes an assembly-level structural stiffness performance subject to maximum stress constraints at component interfaces. The build orientations of each component are modeled by its orientation tensor that avoids numerical instability experienced by the conventional angular representation. A new joint model is introduced at component interfaces, which enables the identification of the interface location, the specification of a distinct material tensor, and imposing maximum stress constraints during optimization. Both 2D and 3D numerical examples are presented to illustrate the effect of the build orientation anisotropy and the component interface behavior on the resulting multicomponent assemblies.

Illusion thermotics with topology optimization

Illusion thermotics, which aims at creating temperature fields to mislead a thermal detector, encompasses many thermal functionalities. It has attracted significant attention due to the increasing number of promising applications. Transformation thermotics and scattering cancellation methods are frequently adopted for illusion devices, though they have inherent defects such as anisotropic thermal properties and complicated fabrication methods. Here, we present a general framework for topology optimization, which can be used to design the structure of an illusion device automatically. The optimized configuration can camouflage the thermal exterior of an illusion device so that it is identical to a reference. The original heat source can drive the virtual heat source despite the structure being simply composed of natural materials. To establish an effective topology optimization model, a volume constraint was imposed on the total volume of iron and a prescribed objective function was used to evaluate the difference between the reference temperature and the temperature field during the topology optimization. The flexibility of the method presented was successfully validated by changing the total volume of iron, the number and locations of the thermal illusions, the types of composite materials, and the original heat sources. Topology optimization is a new and flexible approach for designing thermal metamaterials and metadevices for diverse thermal functionalities and beyond.

A CAD-oriented structural topology optimization method

In this paper, we develop a CAD-oriented topology optimization method incorporating directly the CAD model into the feature-driven optimization (FDO) framework. Practically, any free-form design domain model (FDDM), defined by the CAD model in stereolithography (STL) file format, is first represented by a level set function employing a radial basis function (RBF) interpolation technique. Feature-based topology variation model (TVM) consisting of design and non-design features is then constructed, with the former affording design flexibility whilst the latter preserving the components for engineering requirements. By coupling the FDDM with the TVM via Boolean intersection operation, the topology optimization framework is constructed, and each analysis problem is solved by the finite cell method (FCM). A boundary integral scheme is implemented for efficient sensitivity analysis. It is shown that the developed method shall pose almost no restriction on the expertise of modeling and meshing related to the conventional finite element method. Several examples of increasing complexities are dealt with to demonstrate the effectiveness and the generality of the proposed method.

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.

Topology optimization of 3D continuum structures under geometric self-supporting constraint

The potential of additive manufacturing (AM) in fabricating structurally efficient yet geometrically complicated designs generated from topology optimization is widely recognised. However, various constraints presented in the additive manufacturing process are not directly considered in conventional topology optimization frameworks, potentially impairing the manufacturability of the generated topology. A most noteworthy example is the presence of overhangs, which are downward-facing regions that are not self-supporting. In such areas, traditional AM processes utilise sacrificial supports to act as scaffoldings, which result in additional time, effort and the corresponding cost. Here we present a novel method that can effectively address overhanging features in the bi-directional evolutionary structural optimization framework, creating self-supporting yet structurally efficient designs. By using a simple expression, the overhang problem is formulated as a layer-wise relationship, and elements whose modification does not create overhang are selected as candidates for the element updating scheme. To validate the effectiveness of the algorithm, numerical examples under different design conditions are presented. A comparison of the self-supporting designs against their non-self-supporting counterparts obtained by conventional topology optimization demonstrates their competitiveness in structural performance. Subsequently, physical 3D prints of the examples further prove the effectiveness of the proposed method in creating self-supporting designs.

Explicit topology optimization of novel polyline-based core sandwich structures using surrogate-assisted evolutionary algorithm

Inspired by venation patterns of leaves, the new polyline-based core sandwich structures (PCSSs) are developed in this paper. A novel approach is proposed under explicit topology optimization framework based on surrogate-assisted evolutionary algorithm. Specifically, a multi-component topology description function is defined to characterize the shapes of polyline-based cores, and therefore the dimension of design variables in topology optimization of PCSSs is reduced to only tens of dimensions. To solve the optimization problem, a feedback mechanism-driven surrogate-assisted differential evolution algorithm is proposed, where two search phases, i.e., global and local search phases, are respectively implemented in sequence to achieve a good balance between global exploration and local exploitation. Specifically, the global search phase is utilized to not only explore unknown areas for locating the promising regions effectively, but also improve the accuracies of surrogates in specific promising areas. The local search phase is used to intensively search the local promising regions for accelerating the convergence rate. A feedback mechanism is proposed to drive the adaptive search in the global search phase based on three different kinds of feedback information. Based on the superiorities of constraint satisfactions of PCSSs, a volume domination-based ε-constraint handling method is employed to handle volume and other constraints differently in an appropriate way. Numerical examples are provided to demonstrate the effectiveness and efficiency of the proposed method.

A new data-driven topology optimization framework for structural optimization

The application of structural topology optimization with complex engineering materials is largely hindered due to the complexity in phenomenological or physical constitutive modeling from experimental or computational material data sets. In this paper, we propose a new data-driven topology optimization (DDTO) framework to break through the limitation with the direct usage of discrete material data sets in lieu of constitutive models to describe the material behaviors. This new DDTO framework employs the recently developed data-driven computational mechanics for structural analysis which integrates prescribed material data sets into the computational formulations. Sensitivity analysis is formulated by applying the adjoint method where the tangent modulus of prescribed uniaxial stress-strain data is evaluated by means of moving least square approximation. The validity of the proposed framework is well demonstrated by the truss topology optimization examples. The proposed DDTO framework will provide a great flexibility in structural design for real applications.

A novel application framework for self-supporting topology optimization

This paper presents an application framework that provides a complete process to design an optimized self-supporting structure, ready to be fabricated via additive manufacturing without the usage of additional support structures. Such supports in general have to be created during the fabricating process so that the primary object can be manufactured layer by layer without collapse; this process is very time-consuming and waste of material. The main approach resolves this issue by formulating the self-supporting requirements as an explicit quadratic continuous constraint in a topology optimization problem, or specifically, requiring the number of unsupported elements (in terms of the sum of squares of their densities) to be zero. Under the formulation, the required sensitivity of the self-supporting constraint with respect to the design density can be derived straightforward and is only linearly dependent on the density of the element itself. In addition, a novel discrete convolution operator is particularly designed to detect the unsupported elements. The approach works for cases of general overhang angles, and the produced optimized structures have close target compliance to those of the reference structures obtained without considering the self-supporting constraint, as demonstrated by various 2D and 3D benchmark examples.

Three-dimensional adaptive mesh refinement in stress-constrained topology optimization

Structural optimization software that can produce high-resolution designs optimized for arbitrary cost and constraint functions is essential to solve real-world engineering problems. Such requirements are not easily met due to the large-scale simulations and software engineering they entail. In this paper, we present a large-scale topology optimization framework with adaptive mesh refinement (AMR) applied to stress-constrained problems. AMR allows us to save computational resources by refining regions of the domain to increase the design resolution and simulation accuracy, leaving void regions coarse. We discuss the challenges necessary to resolve such large-scale problems with AMR, namely, the need for a regularization method that works across different mesh resolutions in a parallel environment and efficient iterative solvers. Furthermore, the optimization algorithm needs to be implemented with the same discretization that is used to represent the design field. To show the efficacy and versatility of our framework, we minimize the mass of a three-dimensional L-bracket subject to a maximum stress constraint and maximize the efficiency of a three-dimensional compliant mechanism subject to a maximum stress constraint.

Ambiguous phase assignment of discretized 3D geometries in topology optimization

This paper considers topology optimization approaches where the geometry is described by a level set method and the system’s response is discretized on nonconforming meshes while providing a crisp definition of interface and external boundaries. Since the interface is explicitly tracked, the elements intersected by the interface are divided into sub-elements to which a phase needs to be assigned. Due to loss of information in the discretization of the level set field, certain geometrical configurations allow for ambiguous phase assignment of sub-elements, and thus ambiguous definition of the interface. The study presented here focuses on investigating these topological ambiguities in embedded geometries constructed from discretized level set fields on hexahedral meshes. Three-dimensional problems where several intersection configurations can significantly affect the problem’s topology are considered. This is in contrast to two-dimensional problems where ambiguous topological features exist only in one intersection configuration, and identifying and resolving them is straightforward. A set of rules that resolve these ambiguities for two-phase problems is proposed, and algorithms for their implementations are provided. The influence of these rules on the evolution of the geometry in the optimization process is investigated with linear elastic topology optimization problems. These problems are solved by an explicit level set topology optimization framework that uses the extended finite element method to predict physical responses. This study shows that the choice of a rule to resolve topological features can result in drastically different final geometries. However, for the problems studied in this paper, the performances of the optimized design do not differ.

A new topology optimization framework for stiffness design of beam structures based on the transformable triangular mesh algorithm

Abstract This paper presents an explicit topology optimization method based on the so called Transformable Triangular Mesh (TTM) approach. It improves a general weakness of the traditional explicit approaches in the sense that the genus of initial geometry no longer affects the design result. In the proposed method, we first triangulate the initial geometry and organize it with the halfedge data structure. Then we employ the Laplacian energy controlled mesh deformation algorithm to realize smooth transition from the initial TTM. Furthermore, three geometrical mesh processing techniques (mesh subdivision, mesh split, and mesh refinement) are adopted to realize surface genus change. Respectively, the mesh subdivision is beneficial for the increase of mesh control hot points to obtain more degree of freedoms, the mesh split is able to rip connected triangles such that holes can be generated both inside a mesh and on the border, and the mesh refinement unifies the sizes and shapes of each triangle in the mesh such that no large distortion and self-intersection exist. To enhance the computational efficiency, we realize the above algorithm parallelly using CUDA. Finally, some benchmark examples (a cantilever beam, an MBB beam, and a cantilever beam with a fixed hole) are used to validate the algorithm. By comparing with state-of-the-art algorithms, our method proves to be reliable, accurate, and advanced.