Isogeometric analysis and shape optimization via boundary integral

In this paper, we present a boundary integral based approach to isogeometric analysis and shape optimization.For analysis, it uses the same basis, Non-Uniform Rational B-Spline (NURBS) basis, for both representing object boundary and for approximating physical fields in analysis via a Boundary-Integral-Equation Method (BIEM). We propose the use of boundary points corresponding to Greville abscissae as collocation points. We conducted h-, p- and k-refinement study for linear elasticity and heat conduction problems. Our numerical experiments show that collocation at Greville abscissae leads to overall better convergence and robustness. Replacing rational B-splines with the linear B-Splines as shape functions for approximating solution space in analysis does not yield significant difference in convergence.For shape optimization, it uses NURBS control points to parameterize the boundary shape. A gradient based optimization approach is adopted where analytical sensitivities of how control points affect objective and constraint functions are derived. Two 3D shape optimization examples are demonstrated.Our study finds that the boundary integral based isogeometric analysis and optimization have the following advantages: (1) the NURBS based boundary integral exhibits superior computational advantages over the usual Lagrange polynomials based BIEM on a per degree-of-freedom basis; (2) it bypasses the need for domain parameterization, a bottleneck in current NURBS based volumetric isogeometric analysis and shape optimization; (3) it offers tighter integration of CAD and analysis since both the geometric models for both analysis and optimization are the same NURBS geometry.     

Full analytical sensitivities in NURBS based isogeometric shape optimization

Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.     

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.     

Volumetric mesh generation from T-spline surface representations

A new approach to obtain a volumetric discretization from a T-spline surface representation is presented. A T-spline boundary zone is created beneath the surface, while the core of the model is discretized with Lagrangian elements. T-spline enriched elements are used as an interface between isogeometric and Lagrangian finite elements. The thickness of the T-spline zone and thereby the isogeometric volume fraction can be chosen arbitrarily large such that pure Lagrangian and pure isogeometric discretizations are included. The presented approach combines the advantages of isogeometric elements (accuracy and smoothness) and classical finite elements (simplicity and efficiency).Different heat transfer problems are solved with the finite element method using the presented discretization approach with different isogeometric volume fractions. For suitable applications, the approach leads to a substantial accuracy gain.     

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.     

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.     

Application of isogeometric method to free vibration of Reissner–Mindlin plates with non-conforming multi-patch

The isogeometric method is used to study the free vibration of thick plates based on Mindlin theory. The Non-uniform Rational B-Spline (NURBS) basis functions are employed to build the thick plate’s geometry models and serve as the shape functions for solution field approximation in finite element analysis. The Reissner–Mindlin plates built with multiple NURBS patches are investigated, in which several patches of the model have multi-interface and different patches may share a common point. In order to solve the non-conforming interface problems, Nitsche method is employed to glue different NURBS patches and only refers to the coupling conditions in this work. Various plate shapes, different boundary conditions and several kinds of thickness-span ratios are considered to verify the validity of the presented method. The dimensionless frequencies for different cases are obtained by solving the eigenvalue equation problems and compared with the existing reference solutions or the results calculated by ABAQUS software. Several numerical examples exhibit the effectiveness of the isogeometric approach. It shows that the natural frequencies of the Reissner–Mindlin plate can be successfully predicted by the combination of isogeometric analysis and Nitsche method.     

Isogeometric design sensitivity analysis and experimental validation of nanoscale structures considering surface effects

A continuum-based design sensitivity analysis (DSA) method is developed for nanoscale structures with surface effects. To account for the effects of precise geometry in the response and the design sensitivity analyses, we employ an isogeometric approach which uses the same NURBS basis functions as used to describe the geometry of CAD. A direct differentiation method is employed to obtain the analytical design sensitivity using a generalized Young-Laplace equation with high-order surface effects. Effective material properties with the surface effects for silver nanowires are measured from a three-point bending test using atomic force microscopy (AFM). The diameter and the suspended length of silver nanowires are considered as sizing and shape design variables, respectively. The design sensitivity expressions are derived with respect to the design parameters and validated comparing with the experimental results from the AFM scanning, showing an acceptable agreement.     

Generalized B-splines as a tool in isogeometric analysis

The concept of isogeometric analysis has been proposed in [13], where NURBS are considered as basis of the analysis, thanks to their ability to construct an exact geometric model in several practical applications and to their popularity in commercial CAD systems. In this paper we propose an alternative to the rational model presenting an isogeometric analysis approach based on generalized B-splines. Geometric models exactly represented by generalized B-splines include those generated by NURBS. Moreover, generalized B-splines possess all fundamental properties of algebraic B-splines (and NURBS) including classical refinement processes as h–p–k refinements. Finally, since generalized B-splines are not confined to rational functions, they behave completely similar to algebraic B-splines with respect to differentiation and integration. This seems to be of interest in the treatment of some relevant problems.     

ISOCOMP: Unified geometric and material composition for optimal topology design

In this paper, a unified strategy is developed to simultaneously insert inclusions or holes of regular shape as well as redistribute the material to effect optimal topologies of solids. We demonstrate the unified optimal design strategy through three possible choices of design variables: (1) purely geometrical, (2) purely material, and (3) geometrical-material. We couple the geometrical approach with the topological derivative of the objective function and a condition derived for optimally inserting an infinitesimal ellipsoidal heterogeneity (hole or inclusion) into the structure. The approximations of the geometry, material and behavioral fields are isoparametric (or “isogeometric”) and are composed consistent with the Hierarchical Partition of Unity Field Compositions (HPFC) theory (Rayasam et al., Int J Numer Methods Eng 72(12):1452–1489, 2007). Specifically, analogous to the constructive solid geometry procedure of CAD, the complex material as well as the behavioral field is modeled hierarchically through a series of pair-wise compositions of primitive fields defined on the primitive geometrical domains. The geometrical, material and behavioral approximations are made using Non-Uniform Rational B-Splines (NURBS) basis functions. Thus, the proposed approach seamlessly unifies the explicit representation of boundary shapes with the implicit representations of boundaries arising out of material redistribution, and is termed ISOCOMP, or isoparametric compositions for topology optimization. The methodology is demonstrated first on a set of example problems that increase in complexity of design variable choice culminating in simultaneous optimization of hole location, hole shape and material distribution within the domain. This is followed by a detailed case study involving topology optimization of a bicycle “dropout.”     

T 样条用于计算机辅助设计、分析和制造的 新型表示方法

几何造型主要研究在计算机环境下几何模型的表示与设计,其是计算机辅助设计、工程和制造领 域中重要的基础研究方向。当前工业制造中几何模型表示的标准是非均匀有理 B 样条(NURBS)。然而,由于张量 积拓扑结构,NURBS 在表示复杂几何模型时具有难以克服的局限性。T 样条是一种新型兼容 NURBS 的自由曲面 造型表示技术。由于克服了 NURBS 的多个重要的局限,T 样条自提出后引起了学术界和工业界的极大关注。为 了使国内外同行对 T 样条的发展历程和研究现状有一个较为全面的了解,对其进行系统综述。在广泛文献调研的 基础上,对 T 样条的定义、基础算法,以及在计算机辅助设计、工程和制造中的应用进行归纳总结和详细分析, 重点分析了这些算法的基本思想和原理,比较了其优点和不足。由于工业界对几何表示的精度和效率的要求越来 越高,导致 T 样条的研究仍然不完善,还存在大量亟待解决的问题和可能的发展方向。     

On the numerical implementation of continuous adjoint sensitivity for transient heat conduction problems using an isogeometric approach

A crucial problem of continuous adjoint shape sensitivity analysis is the numerical implementation of its lengthy formulations. In this paper, the numerical implementation of continuous adjoint shape sensitivity analysis is presented for transient heat conduction problems using isogeometric analysis, which can serve as a tutorial guide for beginners. Using the adjoint boundary and loading conditions derived from the design objective and the primary state variable fields, the numerical analysis procedure of the adjoint problem, which is solved backward in time, is demonstrated. Following that, the numerical integration algorithm of the shape sensitivity using a boundary approach is provided. Adjoint shape sensitivity is studied with detailed explanations for two transient heat conduction problems to illustrate the numerical implementation aspects of the continuous adjoint method. These two problems can be used as benchmark problems for future studies.     

质量约束的三维模型建模方法

在当今的智能制造和工业软件设计领域,CAD和CAE是极为重要的技术。但现有的CAD和CAE模型表示方法不统一,在数据交换上需要耗费大量时间,造成了计算资源极大的浪费。非均匀有理B样条(NURBS)模型作为一种兼容CAD和CAE的模型表达方式,以样条曲线为基函数,无须进行交换即可进行等几何分析。本文提出了一种基于质量约束的NURBS体建模方法,将模型质量作为建模时的约束,使构建的模型符合等几何分析的要求。本文以带有复连通域的模型为例进行等几何分析,经过与主流商业软件的对比,最小值相同,最大值误差在10%以内,运行时间减少了4.61%,验证了此方法的正确性。