Strength-based topology optimisation of plastic isotropic von Mises materials

Conventionally, topology optimisation is formulated as a non-linear optimisation problem, where the material is distributed in a manner which maximises the stiffness of the structure. Due to the nature of non-linear, non-convex optimisation problems, a multitude of local optima will exist and the solution will depend on the starting point. Moreover, while stress is an essential consideration in topology optimisation, accounting for the stress locally requires a large number of constraints to be considered in the optimisation problem; therefore, global methods are often deployed to alleviate this with less control of the stress field as a consequence. In the present work, a strength-based formulation with stress-based elements is introduced for plastic isotropic von Mises materials. The formulation results in a convex optimisation problem which ensures that any local optimum is the global optimum, and the problems can be solved efficiently using interior point methods. Four plane stress elements are introduced and several examples illustrate the strength of the convex stress-based formulation including mesh independence, rapid convergence and near-linear time complexity.

     

Deep learning driven real time topology optimisation based on initial stress learning

Topology optimisation can facilitate engineers in proposing efficient and novel conceptual design schemes, but the traditional FEM based optimization demands significant computing power and makes the real time optimization impossible. Based on the convolutional neural network (CNN) method, a new deep learning approximate algorithm for real time topology optimisation is proposed. The algorithm learns from the initial stress (LIS), which is defined as the major principal stress matrix obtained from finite element analysis in the first iteration of classical topology optimisation. The initial major principal stress matrix of the structure is used to replace the load cases and boundary conditions of the structure as independent variables, which can produce topological prediction results with high accuracy based on a relatively small number of samples. Compared with the traditional topology optimisation method, the new method can produce a similar result in real time without repeated iterations. A classic short cantilever problem was used as an example, and the optimized topology of the cantilever structure is predicted successfully by the established approximate algorithm. By comparing the prediction results to the structural optimisation results obtained by the classical topology optimisation method, it is discovered that the two results are highly approximate, which verifies the validity of the established algorithm. Furthermore, a new algorithm evaluation method is proposed to evaluate the effects of using different methods to select samples on the prediction performance of the optimized topology, and the results were promising and concluded in the end.     

Solving stress and compliance constrained volume minimization using anisotropic mesh adaptation,the method of moving asymptotes and a global p-norm

The p-norm often used in stress constrained topology optimisation supposedly mimics a delta function and it is thus characterised by a small length scale and ideally one would also prefer to have the solid-void transition occur over a small length scale, since the material in this transition does not have a clear physical interpretation. We propose to resolve these small length scales using anisotropic mesh adaptation. We use the method of moving asymptotes with interpolation of sensitivities, asymptotes and design variables between iterations. We demonstrate this combination for the portal and L-bracket problems with p=10, and we are able to investigate mesh dependence. Finally, we suggest relaxing the L-bracket problem statement by introducing a rounded corner.     

Combined shape and topology optimization for minimization of maximal von Mises stress

This work shows that a combined shape and topology optimization method can produce optimal 2D designs with minimal stress subject to a volume constraint. The method represents the surface explicitly and discretizes the domain into a simplicial complex which adapts both structural shape and topology. By performing repeated topology and shape optimizations and adaptive mesh updates, we can minimize the maximum von Mises stress using the p-norm stress measure with p-values as high as 30, provided that the stress is calculated with sufficient accuracy.     

A simple alternative formulation for structural optimisation with dynamic and buckling objectives

Structural topology optimisation has mainly been applied to strength and stiffness objectives, due to the ease of calculating the sensitivities for such problems. In contrast, dynamic and buckling objectives require time consuming central difference schemes, or inefficient non-gradient algorithms, for calculation of the sensitivities. Further, soft-kill algorithms suffer from numerous numerical issues, such as localised artificial modes and mode switching. This has resulted in little focus on structural topology optimisation for dynamic and buckling objectives. In this work it is found that nominal stress contours can be derived from applying the vibration and buckling mode shapes as displacement fields, defined as the dynamic and buckling von Mises stress, respectively. This paper shows that there is an equivalence between the dynamic von Mises stress and the frequency sensitivity numbers for element removal and addition in bidirectional evolutionary structural optimisation. Likewise, it was found that the contours of buckling von Mises stress and buckling sensitivity numbers are analogous; therefore, an equivalence is shown for element removal and addition. The examples demonstrate consistent resulting topologies from the two different formulations for both dynamic and buckling criteria. This article aims to develop a simple alternative, based on visual correlation with a mathematical verification, for topology optimisation with dynamic and buckling criteria.     

基于凸凹性的网格细分综合优化

实际工程中希望表示物体的三角形网格形状优良,同时拓扑逼近真实曲面。但是对非均匀离散点云重建得到的网格进行优化时,这两个标准常常是相互矛盾的。该文针对在实际工程中遇见的这个问题,提出一种结合全局特征以及局部特性的细分算法。该算法避免了一般细分方法对凹区域处理出现的折叠现象,可以获取三角形形状和空间拓扑的综合优化解。最后通过对于工程应用实例的细分计算,得到了与原始网格拓扑一致,但更逼近真实曲面的细分优化网格,表明了所提出简化算法的有效性。     

Topology optimization using an explicit interface representation

We introduce the Deformable Simplicial Complex method to topology optimization as a way to represent the interface explicitly yet being able to handle topology changes. Topology changes are handled by a series of mesh operations, which also ensures a well-formed mesh. The same mesh is therefore used for both finite element calculations and shape representation. In addition, the approach unifies shape and topology optimization in a complementary optimization strategy. The shape is optimized on the basis of the gradient-based optimization algorithm MMA whereas holes are introduced using topological derivatives. The presented method is tested on two standard minimum compliance problems which demonstrates that it is both simple to apply, robust and efficient.     

Effective optimisation of continuum topologies through a multi-GA system

The success of GA-based topology optimisation methods for continuum structures has been limited. Though a number of methods exist, the results are far from comparable to solutions from more established methods in terms of quality and fidelity. This had led many structural optimisation researchers to dismiss the GA as a viable method for topology optimisation of continuum structures. The authors believe however that the deficiencies of previous applications lie not in the GA but in the manner in which it was applied. Based on insights gained from these applications, a new GA-based method for the continuum topology optimisation problem is herein presented which yields high-fidelity solutions comparable to those produced through more well-known and established methods. The method is centred on a number of novel concepts, most notably the simultaneous use of multiple genetic algorithms, and the use of a high-level string coding based on structural response information. In this work, the basis and methodology of the multi-GA approach are explained, and key algorithms detailed. The method was also applied to a number of structural problems to show its efficacy, and the results presented and discussed.     

Interior value extrapolation: a new method for stress evaluation during topology optimization

This article presents a new method for evaluating stresses in the jagged structures that arise when using a fixed finite element mesh to optimize the topology of a structure. The new method, Interior Value Extrapolation, IVE, exploits the fact that in the interior of the structure, the stresses calculated by the finite element method are more accurate than at the boundary. The jagged nature of the mesh makes stresses at the boundary oscillate. Therefore, stresses at the boundary are instead extrapolated from results in the interior, resulting in a more stable and accurate stress measure. A restriction method in the form of a non linear density filter is also proposed, tailored to be used in conjunction with the new stress evaluation method. The new method is evaluated for accuracy using example geometries, for which the stresses are known. It is shown that IVE improves the accuracy of the stress calculation. Optimization examples are thereafter solved with and without IVE, and the results are discussed. It is shown that the change in stress evaluation can in fact cause changes in the solution of a typical stress minimization problem.     

Genetic algorithm-based optimisation of load-balanced routing for AMI with wireless mesh networks

The advanced metering infrastructure (AMI) in a smart grid contains hardware, software, and other electronic components connected through a communication infrastructure. AMI transfers meter-reading data between a group of smart meters and a utility centre. Herein, a wireless mesh network (WMN) with a random mesh topology is used to deploy the AMI communication network. In a WMN, paths are identified using a hybrid wireless mesh routing protocol (HWMP) with a load balancing feature called load aware-HWMP (LA-HWMP). These paths reduce the demand on links with a minimal air time metric; however, the delay in the data transmission of certain smart meters is high, given the large number of retransmissions caused by packet drop. To avert this problem and enhance the end-to-end delay, a genetic algorithm is applied on the LA-HWMP to obtain the optimal path. The optimisation process will result in the selection of paths with minimal delay. The genetic algorithm is developed with a rank-based selection, a two-point crossover, and a random reset mutation with a repair function to eliminate duplicate entries. The proposed method is compared with the HWMP, the LA-HWMP, and a state-of-the-art method that uses a combination of the ant colony algorithm and simulated annealing (ACA-SA) for AMI networks of different sizes. The obtained results show that the path identified by the proposed method yields a shorter delay and higher throughput than paths identified using the other methods.     

A Colour Interpolation Scheme for Topologically Unrestricted Gradient Meshes

Gradient meshes are a 2D vector graphics primitive where colour is interpolated between mesh vertices. The current implementations of gradient meshes are restricted to rectangular mesh topology. Our new interpolation method relaxes this restriction by supporting arbitrary manifold topology of the input gradient mesh. Our method is based on the Catmull‐Clark subdivision scheme, which is well‐known to support arbitrary mesh topology in 3D. We adapt this scheme to support gradient mesh colour interpolation, adding extensions to handle interpolation of colours of the control points, interpolation only inside the given colour space and emulation of gradient constraints seen in related closed‐form solutions. These extensions make subdivision a viable option for interpolating arbitrary‐topology gradient meshes for 2D vector graphics.     

Topology-preserving simplification of 2D nonmanifold meshes with embedded structures

Mesh simplification has received tremendous attention over the years. Most of the previous work in this area deals with a proper choice of error measures to guide the simplification. Preserving the topological characteristics of the mesh and possibly of data attached to the mesh is a more recent topic and the subject of this paper. We introduce a new topology-preserving simplification algorithm for triangular meshes, possibly nonmanifold, with embedded polylines. In this context, embedded means that the edges of the polylines are also edges of the mesh. The paper introduces a robust test to detect if the collapse of an edge in the mesh modifies either the topology of the mesh or the topology of the embedded polylines. This validity test is derived using combinatorial topology results. More precisely, we define a so-called extended complex from the input mesh and the embedded polylines. We show that if an edge collapse of the mesh preserves the topology of this extended complex, then it also preserves both the topology of the mesh and the embedded polylines. Our validity test can be used for any 2-complex mesh, including nonmanifold triangular meshes, and can be combined with any previously introduced error measure. Implementation of this validity test is described. We demonstrate the power and versatility of our method with scientific data sets from neuroscience, geology, and CAD/CAM models from mechanical engineering.     

Adaptive mesh refinement in stress-constrained topology optimization

We present a topology structural optimization framework with adaptive mesh refinement and stress-constraints. Finite element approximation and geometry representation benefit from such refinement by enabling more accurate stress field predictions and greater resolution of the optimal structural boundaries. We combine a volume fraction filter to impose a minimum design feature size, the RAMP penalization to generate “black-and-white designs” and a RAMP-like stress definition to resolve the “stress singularity problem.” Regions with stress concentrations dominate the optimized design. As such, rigorous simulations are required to accurately approximate the stress field. To achieve this goal, we invoke a threshold operation and mesh refinement during the optimization. We do so in an optimal fashion, by applying adaptive mesh refinement techniques that use error indicators to refine and coarsen the mesh as needed. In this way, we obtain more accurate simulations and greater resolution of the design domain. We present results in two dimensions to demonstrate the efficiency of our method.     

An analysis of simplex shape measures for anisotropic meshes

This paper analyses and generalises several simplex shape measures documented in the literature and currently used for mesh adaptation and mesh optimisation. It first summarises important properties of simplices and their degeneration in Euclidean space. Different simplex shape measures are then defined and validated according to a validity criterion. The shape measures are generalised to Riemannian spaces in order to extend their use to anisotropic meshes. They are then analysed and compared using complementary approaches: a visualisation method helping to show their regularity, some theoretical relations establishing their equivalence, and a discussion on the evaluation of the global quality of a mesh. Conclusions are drawn on the choice of a simplex shape measure to guide mesh optimisation.     

基于ANSYS二次开发的结构拓扑优化

将先进的结构拓扑优化思想与成熟的有限元分析软件ANSYS相结合,基于ANSYS二次开发语言APDL和UIDL编制了结构拓扑优化程序,解决了ANSYS自带的拓扑优化模块中单元类型受限及不能应用于桁架结构优化的问题,充分利用ANSYS丰富的单元类型、强大的计算和后处理能力,有利于促进结构拓扑优化的应用和研究,拓宽ANSYS软件在结构拓扑优化方面的应用领域。     

Shape optimisation of preform design for precision close-die forging

Preform design is an essential stage in forging especially for parts with complex shapes. In this paper, based on the evolutionary structural optimisation (ESO) concept, a topological optimisation method is developed for preform design. In this method, a new criterion for element elimination and addition on the workpiece boundary surfaces is proposed to optimise material distribution. To improve the quality of the boundary after element elimination, a boundary smoothing technique is developed using B-spline curve approximation. The developed methods are programmed using C# code and integrated with DEFORM 2D software package. Two 2D case problems including forging of an aerofoil shape and forging of rail wheel are evaluated using the developed method. The results suggest that the developed topology optimisation method is an efficient approach for preform design optimisation.     

Fast Mesh Reconstruction from Single View Based on GCN and Topology Modification

3D reconstruction based on single view aims to reconstruct the entire 3D shape of an object from one perspective. When existing methods reconstruct the mesh surface of complex objects, the surface details are difficult to predict and the reconstruction visual effect is poor because the mesh representation is not easily integrated into the deep learning framework; the 3D topology is easily limited by predefined templates and inflexible, and unnecessary mesh self-intersections and connections will be generated when reconstructing complex topology, thus destroying the surface details; the training of the reconstruction network is limited by the large amount of information attached to the mesh vertices, and the training time of the reconstructed network is too long. In this paper, we propose a method for fast mesh reconstruction from single view based on Graph Convolutional Network (GCN) and topology modification. We use GCN to ensure the generation of high-quality mesh surfaces and use topology modification to improve the flexibility of the topology. Meanwhile, a feature fusion method is proposed to make full use of the features of each stage of the image hierarchically. We use 3D open dataset ShapeNet to train our network and add a new weight parameter to speed up the training process. Extensive experiments demonstrate that our method can not only reconstruct object meshes on complex topological surfaces, but also has better qualitative and quantitative results.     

有限元网格图拓扑分析

依据图论的方法对有限元网格图进行了拓朴分析，讨论了单元节点间的相关性，提出了构造单元网格节点拓扑阵和组集整体网格节点拓扑阵的方法。这是一种新的有限元网格图自动生成方法，简洁明快，具有良好的通用性。     

HexBox: Interactive Box Modeling of Hexahedral Meshes

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid-based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques.     

Generalized incremental frequency method for topological designof continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency

The application of multidisciplinary design optimisation is mostly confined to bi-disciplinary systems such as fluid-structure interaction problems. High fidelity models of three disciplines involving electromagnetic-thermal-structural designs are rare. Here, the multidisciplinary optimisation of such a design is presented. The device comprises a C-shaped iron core and a single coil. The problem is decomposed using a monolithic multidisciplinary feasible architecture. The multidisciplinary analyses involve a single three-dimensional finite element mesh for transient non-linear electromagnetic, non-linear-static thermal, and linear-static structural models. During each multidisciplinary iteration the mesh is linearly morphed. A gradient based optimisation algorithm in combination with a multi-start routine is applied to the constrained mass minimisation problem. Multidisciplinary feasibility is ensured by convergence of a single coupling parameter i.e. air-gap deformation. In conclusion, some multidisciplinary optimisation, analyses, and decomposition considerations are discussed.