Topology optimization for hybrid additive-subtractive manufacturing

Hybrid additive-subtractive manufacturing is gaining popularity by making full use of geometry complexity produced by additive manufacturing and dimensional accuracy derived from subtractive machining. Part design for this hybrid manufacturing approach has been done by trial-and-error, and no dedicated design methodology exists for this manufacturing approach. To address this issue, this work presents a topology optimization method for hybrid additive and subtractive manufacturing. To be specific, the boundary segments of the input design domain are categorized into two types: (i) Freeform boundary segments freely evolve through the casting SIMP method, and (ii) shape preserved boundary segments suppress the freeform evolvement and are composed of machining features through a feature fitting algorithm. Given the manufacturing strategy, the topology design is produced through additive manufacturing and the shape preserved boundary segments will be processed by post-machining. This novel topology optimization algorithm is developed under a unified SIMP and level set framework. The effectiveness of the algorithm is proved through a few numerical case studies.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.     

基于T-Spline的全自动几何拓扑修复方法

从高质量曲面网格生成的需求出发,提出了一种基于T-Spline的全自动几何拓扑修复方法.本文方法创新性主要可归纳为:1）对原有计算机辅助设计（Computer aided design,CAD）几何模型不进行任何修改保留其本真,自动识别CAD几何模型中常见不必要的几何特征,成功解决了CAD几何模型中存在的几何瑕疵,如短边、窄面、退化边、退化面、非连续光滑边界及尖锐特征等,利用新生成的"虚边"、"虚面"处理几何瑕疵,同时通过虚拓扑重构CAD几何模型的B-Rep;2）开发了一套CAD/CAE集成系统,统一了几何模型与计算分析模型,实现计算机辅助工程（Computer aided engineering,CAE）与CAD两者的无缝集成,所有拓扑修复操作及后续CAE分析计算均在同一环境下进行,避免了几何模型在CAE与CAD系统间进行转换时造成的数据丢失.该方法能够对复杂实体实现全自动几何拓扑修复及网格生成,实验表明,在保证不失真的前提下,修复后的几何模型能够生成质量良好的网格且能降低网格的生成规模,验证了本文方法的实用性和有效性,以满足工程实际分析的需要.     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

A CAD-based design parameterization for shape optimization of elastic solids

In this paper a CAD-based design sensitivity analysis (DSA) and optimization method using Pro/ENGINEER for shape design of structural components is presented. The CAD-based design model is critically important for multidisciplinary shape design optimization. Only when each discipline can compute the design sensitivity coefficients of the CAD-based design model, can a true multidisciplinary what-if study, trade-off analysis, and design optimization be carried out. The proposed method will allow the design engineer to compute design sensitivity coefficients of structural performance measures such. as stress and displacement, evaluated using existing finite element analysis (FEA) tools, both h- and p-versions, with respect to design variables defined in the parameterized CAD model. The proposed method consists of (i) a CAD-based design parameterization technique that ties the structural DSA and optimization to a CAD tool; (ii) a design velocity field computation that defines material point movement due to design change in CAD geometry, satisfies linearity and regularity requirements, and supports both hand p-version FEA meshed using existing mesh generators; and (iii) a design optimization method that supports structural geometric and finite element model updates in Pro/ENGINEER during the optimization process.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.     

Shape disassembly using generating merging probability

Computer integrated manufacturing uses computer technology to integrate a manufacturing system through a man-machine interface that fills the gap between manual operation and machine processes. It is clear that a computer vision-based man-machine interface makes a fully automated system possible. The basic challenge of a vision-based interface is how to extract information from digitized images and convert it to machine-friendly knowledge. To extract information, then, it often end up to the problem of shape decomposition. This paper proposes an new approach in decomposing compound shapes without prior knowledge of the scene. The proposed algorithm exploits the fact that planar shapes can be completely described by contour segments, and can be decomposed at their maximum concavity into simpler objects. To reduce spurious decomposition, the decomposed segments are merged into groups by analyzing and utilizing the merging hypotheses. The algorithm calculates the linking possibility by weighting the angular differentiation between two segments. The techniques are implemented and are applied to other partial shape matching problems for clustering purposes.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

Shape evolution with structural and topological changes usingblending

This paper describes a framework for the estimation of shape from sparse or incomplete range data. It uses a shape representation called blending, which allows for the geometric combination of shapes into a unified model - selected regions of the component shapes are cut-out and glued together. Estimation of shape by this representation is realized using a physics-based framework, and it also includes a process for deciding how to adapt the structure and topology of the model to improve the fit. The blending representation helps avoid abrupt changes in model geometry during fitting by allowing the smooth evolution of the shape, which improves the robustness of the technique. We demonstrate this framework with a series of experiments showing its ability to automatically extract structured representations from range data given both structurally and topologically complex objects     

Segmentation of ultrasound images of the carotid using RANSAC and cubic splines

A new algorithm is proposed for the semi-automatic segmentation of the near-end and the far-end adventitia boundary of the common carotid artery in ultrasound images. It uses the random sample consensus method to estimate the most significant cubic splines fitting the edge map of a longitudinal section. The consensus of the geometric model (a spline) is evaluated through a new gain function, which integrates the responses to different discriminating features of the carotid boundary: the proximity of the geometric model to any edge or to valley shaped edges; the consistency between the orientation of the normal to the geometric model and the intensity gradient; and the distance to a rough estimate of the lumen boundary.A set of 50 longitudinal B-mode images of the common carotid and their manual segmentations performed by two medical experts were used to assess the performance of the method. The image set was taken from 25 different subjects, most of them having plaques of different classes (class II to class IV), sizes and shapes.The quantitative evaluation showed promising results, having detection errors similar to the ones observed in manual segmentations for 95% of the far-end boundaries and 73% of the near-end boundaries.     

融合语义与几何特征的人体模型结构分割*

针对现有三维人体模型形状分析方法存在人工干预及对姿势依赖的问题,提出一种融合语义与几何特征的三维人体形状分析方法。首先,基于模型表面测地线距离以及内部空间体积特征的度量,提出了基于骨架树的结构检测方法;其次,基于人体测量学先验语义知识,进一步提炼模型的层次结构。该方法能有效的提取不同姿态人体模型的结构特征,并实现基于语义的模型分割,一系列实验结果验证了该方法的高效性与鲁棒性。     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

An open platform of shape design optimization for shell structure

A general platform built on a computer-aided design (CAD) system is developed for parameterized shape design optimization of shell structure. Within the platform, parameterized surface modeling and computer-aided engineering (CAE) applications are embedded and seamlessly integrated with the CAD system through its application programming interface (API). Firstly, instead of the CAD system inherent surface modeling, a parameterized surface modeling for shell structure is fulfilled through integrating with parametric solid modeling of the CAD system. Thus, any dimensions for parametric solid modeling can be used to control shape modification of shell structure and serve as design variables for shape design optimization. Secondly, seamless integration of geometry modeling and finite-element modeling for shell structure is implemented. Finally, with integrated procedures of finite-element analysis and optimization algorithms, a general platform for parameterized shape optimization of shell structure is realized. Numerical examples are presented, and the results validate the effectiveness and efficiency of the platform. A shorten version of this paper was presented to the 7th World Congress of Computation Mechanics (WCCM 2006), July 16–22, 2006, Los Angeles, CA, USA.     

Automation in the design of EDM electrodes

Plastic parts, especially those designed for consumer electronics products, are becoming more compact and freeform in shape. This requires the construction of compact and delicate geometric features in the injection moulds that produce these parts. It is now common practice to employ electric discharge machining (EDM) whenever the conventional machining process fails to machine such delicate features. Although CAD/CAM systems are widely used in mould design and manufacturing, and specific commercial CAD/CAM systems that are customized for injection mould applications are also available, there are still no intelligent CAD tools that address the specific requirements of EDM electrode design. The aim of this research is to develop an intelligent CAD tool that supports EDM electrode design. This CAD tool uses a new technique that recursively splits the EDM region into sub-regions until machinable electrodes can be constructed. The CAD tool has been implemented and integrated into a commercial CAD/CAM system, and 40 different real designs has been used to test its capability in handling a wide variety of geometric shapes. Performance data show that the efficiency of the design process can be improved by at least 50%, with a potential to achieve 85% through improved implementation and further research.