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.     

Isogeometric topological shape optimization using dual evolution with boundary integral equation and level sets

An isogeometric topological shape optimization method is developed, using a dual evolution of NURBS curves and level sets; the NURBS curves feature the exact representation of geometry and the level sets help to detect and guide the topological variation of NURBS curves. The implicit geometry by the level sets is transformed into the parametric NURBS curves by minimizing the difference of velocity fields in both representations. A gradient-based optimization problem is formulated, based on the evolution of the NURBS curves. The control points of NURBS curves are taken as design variables. The necessary response and design sensitivity are computed by an isogeometric boundary integral equation method (BIEM) using the NURBS curves. The design sensitivity is obtained on fixed grids and utilized as the velocity to update the Hamilton–Jacobi equation for the level sets. To obtain the whole velocity field on the fixed grids, an interpolation and velocity extension scheme are employed. The developed method provides accurate response and enhanced sensitivity using isogeometric BIEM. Also, additional post-processing is not required to communicate with CAD systems since the optimal design is represented as NURBS curves. Numerical examples demonstrate the accuracy of design sensitivity on fixed grids and the feasibility of shape and topological optimization.     

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

An analytical representation of conformal mapping for genus-zero implicit surfaces and its application to surface shape similarity assessment

This paper develops an analytical representation of conformal mapping for genus-zero implicit surfaces based on algebraic polynomial functions, and its application to surface shape similarity assessment. Generally, the conformal mapping often works as a tool of planar or spherical parameterization for triangle mesh surfaces. It is further exploited for implicit surface matching in this study. The method begins with discretizing one implicit surface by triangle mesh, where a discrete harmonic energy model related to both the mesh and the other implicit surface is established based on a polynomial-function mapping. Then both the zero-center constraint and the landmark constraints are added to the model to ensure the uniqueness of mapping result with the Möbius transformation. By searching optimal polynomial coefficients with the Lagrange–Newton method, the analytical representation of conformal mapping is obtained, which reveals all global and continuous one-to-one correspondent point pairs between two implicit surfaces. Finally, a shape similarity assessment index for (two) implicit surfaces is proposed through calculating the differences of all the shape index values among those corresponding points. The proposed analytical representation method of conformal mapping and the shape assessment index are both verified by the simulation cases for the closed genus-zero implicit surfaces. Experimental results show that the method is effective for genus-zero implicit surfaces, which will offer a new way for object retrieval and manufactured surface inspection.     

Design sensitivity analysis with isoparametric shell elements

This paper deals with design sensitivity calculation by the direct differentiation method for isoparametric curved shell elements. Sensitivity parameters include geometric variables which influence the size and the shape of a structure, as well as the shell thickness. The influence of design variables, therefore, may be separated into two distinct contributions. The parametric mapping within an element, as well as the influence of geometric variables on the orientation of an element in space, is accounted for by the sensitivity calculation of geometric variables, and efficient formulations of sensitivity calculation are derived for the element stiffness, the geometric stiffness and the mass matrices. The methods presented here are applied to the sensitivity calculations of displacement, stress, buckling stress and natural frequency of typical basic examples such as a square plate and a cylindrical shell. The numerical results are compared with the theoretical solutions and finite difference values.     

Structural application of a shape optimization method based on a genetic algorithm

This paper presents a structural application of a shape optimization method based on a Genetic Algorithm (GA). The method produces a sequence of fixed-distance step-wise movements of the boundary nodes of a finite element model to derive optimal shapes from an arbitrary initial design space. The GA is used to find the optimal or near-optimal combination of boundary nodes to be moved for a given step movement. The GA uses both basic and advanced operators. For illustrative purposes, the method has been applied to structural shape-optimization. The shape-optimization methodology presented allows local optimization, where only crucial parts of a structure are optimized as well as global shape-optimization which involves finding the optimal shape of the structure as a whole for a given environment as described by its loading and freedom conditions. Material can be removed or added to reach the optimal shape. Two examples of structural shape optimization are included showing local and global optimization through material removal and addition. Received October 14, 1999     

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.     

Introducing the sequential linear programming level-set method for topology optimization

This paper introduces an approach to level-set topology optimization that can handle multiple constraints and simultaneously optimize non-level-set design variables. The key features of the new method are discretized boundary integrals to estimate function changes and the formulation of an optimization sub-problem to attain the velocity function. The sub-problem is solved using sequential linear programming (SLP) and the new method is called the SLP level-set method. The new approach is developed in the context of the Hamilton-Jacobi type level-set method, where shape derivatives are employed to optimize a structure represented by an implicit level-set function. This approach is sometimes referred to as the conventional level-set method. The SLP level-set method is demonstrated via a range of problems that include volume, compliance, eigenvalue and displacement constraints and simultaneous optimization of non-level-set design variables.     

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 new method for T-spline parameterization of complex 2D geometries

We present a new strategy, based on the idea of the meccano method and a novel T-mesh optimization procedure, to construct a T-spline parameterization of 2D geometries for the application of isogeometric analysis. The proposed method only demands a boundary representation of the geometry as input data. The algorithm obtains, as a result, high quality parametric transformation between 2D objects and the parametric domain, the unit square. First, we define a parametric mapping between the input boundary of the object and the boundary of the parametric domain. Then, we build a T-mesh adapted to the geometric singularities of the domain to preserve the features of the object boundary with a desired tolerance. The key of the method lies in defining an isomorphic transformation between the parametric and physical T-mesh finding the optimal position of the interior nodes by applying a new T-mesh untangling and smoothing procedure. Bivariate T-spline representation is calculated by imposing the interpolation conditions on points sited both in the interior and on the boundary of the geometry. The efficacy of the proposed technique is shown in several examples. Also we present some results of the application of isogeometric analysis in a geometry parameterized with this technique.     

Improving accuracy of high-order discontinuous Galerkin method for time-domain electromagnetics on curvilinear domains

The paper discusses high-order geometrical mapping for handling curvilinear geometries in high-accuracy discontinuous Galerkin simulations for time-domain Maxwell problems. The proposed geometrical mapping is based on a quadratic representation of the curved boundary and on the adaptation of the nodal points inside each curved element. With high-order mapping, numerical fluxes along curved boundaries are computed much more accurately due to the accurate representation of the computational domain. Numerical experiments for two-dimensional and three-dimensional propagation problems demonstrate the applicability and benefits of the proposed high-order geometrical mapping for simulations involving curved domains.     

3D head model classification by evolutionary optimization of the Extended Gaussian Image representation

Classification of 3D head models based on their shape attributes for subsequent indexing and retrieval are important in many applications, as in hierarchical content-based retrieval of these head models for virtual scene composition, and the automatic annotation of these characters in such scenes. While simple feature representations are preferred for more efficient classification operations, these features may not be adequate for distinguishing between the subtly different head model classes. In view of these, we propose an optimization approach based on genetic algorithm (GA) where the original model representation is transformed in such a way that the classification rate is significantly enhanced while retaining the efficiency and simplicity of the original representation. Specifically, based on the Extended Gaussian Image (EGI) representation for 3D models which summarizes the surface normal orientation statistics, we consider these orientations as random variables, and proceed to search for an optimal transformation for these variables based on genetic optimization. The resulting transformed distributions for these random variables are then used as the modified classifier inputs. Experiments have shown that the optimized transformation results in a significant improvement in classification results for a large variety of class structures. More importantly, the transformation can be indirectly realized by bin removal and bin count merging in the original histogram, thus retaining the advantage of the original EGI representation.     

Numerical shape optimization based on meshless method and stochastic optimization technique

This paper puts forward a newer approach for structural shape optimization by combining a meshless method (MM), i.e. element-free Galerkin (EFG) method, with swarm intelligence (SI)-based stochastic ‘zero-order’ search technique, i.e. artificial bee colony (ABC), for 2D linear elastic problems. The proposed combination is extremely beneficial in structural shape optimization because MM, when used for structural analysis in shape optimization, eliminates inherent issues of well-known grid-based numerical techniques (i.e. FEM) such as mesh distortion and subsequent remeshing while handling large shape changes, poor accuracy due to discontinuous secondary field variables across element boundaries needing costly post-processing techniques and grid optimization to minimize computational errors. Population-based stochastic optimization technique such as ABC eliminates computational burden, complexity and errors associated with design sensitivity analysis. For design boundary representation, Akima spline interpolation has been used in the present work owing to its enhanced stability and smoothness over cubic spline. The effectiveness, validity and performance of the proposed technique are established through numerical examples of cantilever beam and fillet geometry in 2D linear elasticity for shape optimization with behavior constraints on displacement and von Mises stress. For both these problems, influence of a number of design variables in shape optimization has also been investigated.     

基于整数维约束的分维形状映射生成方法

把整数维图形研究方法与分数维图形研究方法相结合，提出一种基于整数维规则几何形状的约束的分维形状映射生成方法，有于描述自然界和工程中出现的且具有特定基本形状趋势的随机现象和随机过程，首先，任意选取一个具有调配函数的自由形状构造一个有序参数空间，确定有序参数的取值规律和区域，把有序参数空间与自适应神经网络，随机性相结合，构造一个随机离散参数空间，并建立起有序参数与随机离散参数之间的参数对应关系，最后，通过有序与无序的参数对应关系，建立一个独立于任意规则几何形状的统一的分形映射关系，对相应整数维的任意规则参数几何形状分形映射，生成宏观形态趋势可预见和可控制的分维形状，该提出的方法适有于任意参数几何形状的分形映射，生成分形图形，且方法简明，易于实现。     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

Boundary finding with parametrically deformable models

Segmentation using boundary finding is enhanced both by considering the boundary as a whole and by using model-based global shape information. The authors apply flexible constraints, in the form of a probabilistic deformable model, to the problem of segmenting natural 2-D objects whose diversity and irregularity of shape make them poorly represented in terms of fixed features or form. The parametric model is based on the elliptic Fourier decomposition of the boundary. Probability distributions on the parameters of the representation bias the model to a particular overall shape while allowing for deformations. Boundary finding is formulated as an optimization problem using a maximum a posteriori objective function. Results of the method applied to real and synthetic images are presented, including an evaluation of the dependence of the method on prior information and image quality