On regularization in parameter-free shape optimization

This contribution is concerned with a parameter-free approach to computational shape optimization of mechanically-loaded structures. Thereby the term ’parameter-free’ refers to approaches in shape optimization in which the design variables are not derived from an existing CAD-parametrization of the model geometry but rather from its finite element discretization. One of the major challenges in using this type of approach is the avoidance of oscillating boundaries in the optimal design trials. This difficulty is mainly attributed to a lack of smoothness of the objective sensitivities and the relatively high number of design variables within the parameter-free regime. To compensate for these deficiencies, Azegami introduced the concept of the so-called traction method, in which the actual design update is deduced from the deformation of a fictitious continuum that is loaded in proportion to the negative shape gradient. We investigate a discrete variant of the traction method, in which the design sensitivities are computed with respect to variations of the design nodes for a given finite element mesh rather than on the abstract level by means of the speed method. Moreover, the design update process is accompanied by adaptive mesh refinement based on discrete material residual forces. Therein, we consider radaptive node relocation as well as hadaptive mesh refinement. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An isogeometric boundary element approach for topology optimization using the level set method

Among the various types of structural optimization, topology has been occupying a prominent place over the last decades. It is considered the most versatile because it allows structural geometry to be determined taking into account only loading and fixing constraints. This technique is extremely useful in the design phase, which requires increasingly complex computational modeling. Modern geometric modeling techniques are increasingly focused on the use of NURBS basis functions. Consequently, it seems natural that topology optimization techniques also use this basis in order to improve computational performance. In this paper, we propose a way to integrate the isogeometric boundary techniques to topology optimization through the level set function. The proposed coupling occurs by describing the normal velocity field from the level set equation as a function of the normal shape sensitivity. This process is not well behaved in general, so some regularization technique needs to be specified. Limiting to plane linear elasticity cases, the numerical investigations proposed in this study indicate that this type of coupling allows to obtain results congruent with the current literature. Moreover, the additional computational costs are small compared to classical techniques, which makes their advantage for optimization purposes evident, particularly for boundary element method practitioners.     

Development of an inverse isogeometric methodology and its application in sheet metal forming process

This paper proposes an inverse isogeometric analysis to estimate the blank and predict the strain distribution in sheet metal forming processes. In this study, the same NURBS basis functions are used for drawing a final part and analysis of the forming process. In other words, this approach requires only one modeling and analysis representation, in contrast to inverse FEM. This model deals with minimization of potential energy, deformation theory of plasticity, and infinitesimal deformation relations with considering a new non-uniform friction model. One advantage of the presented methodology is that the governing equations are solved in two-dimensional space without concerning about pre-estimation results. As a result, the convergence is guaranteed and the computation time decreases significantly which is important at the initial stages of design. Furthermore, by employing this model at the forming design stage, the effects of changing the final part geometry and material property can be simultaneously observed on the formability of the part. Moreover, the effects of isogeometric element size can be automatically studied on the solution accuracy. The capability of this method is demonstrated by presenting three examples including blank estimation of cylindrical cup, square box, and weld line movement in forming of tailor welded blanks. The results obtained by the presented model and those obtained by the forward FEM reveal reasonable accuracy with decreased computational costs.     

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.     

Material optimization of tri-directional functionally graded plates by using deep neural network and isogeometric multimesh design approach

The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method.     

Application of domain decomposition methods to isogeometric analysis

During the past decade various new spatial discretization techniques have been developed. In particular, the usage of NURBS based shape functions, well known to the CAD community, has been adapted to finite element technology. In the present work we use the mortar finite element method for the coupling of nonconforming discretized sub-domains in the framework of nonlinear elasticity. We show that the method can be applied to isogeometric analysis with little effort, once the framework of NURBS based shape functions has been implemented. Furthermore, a specific coordinate augmentation technique allows the design of an energy-momentum scheme for the constrained mechanical system under consideration. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the multi-component layout design with inertial force

The purpose of this paper is to introduce inertial forces into the proposed integrated layout optimization method designing the multi-component systems. Considering a complex packing system for which several components will be placed in a container of specific shape, the aim of the design procedure is to find the optimal location and orientation of each component, as well as the configuration of the structure that supports and interconnects the components. On the one hand, the Finite-circle Method (FCM) is used to avoid the components overlaps, and also overlaps between components and the design domain boundaries. One the other hand, the optimal material layout of the supporting structure in the design domain is designed by topology optimization. A consistent material interpolation scheme between element stiffness and inertial load is presented to avoid the singularity of localized deformation due to the presence of design dependent inertial loading when the element stiffness and the involved inertial load are weakened with the element material removal. The tested numerical example shows the proposed methods extend the actual concept of topology optimization and are efficient to generate reasonable design patterns.     

Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations

Collocation is based on the discretization of the strong form of the underlying partial differential equations, which requires basis functions of sufficient order and smoothness. Consequently, the use of isogeometric analysis (IGA) for collocation suggests itself, since splines can be readily adjusted to any order in polynomial degree and continuity required by the differential operators. In addition, they can be generated for domains of arbitrary geometric and topological complexity, directly linked to and fully supported by CAD technology. The major advantage of isogeometric collocation over Galerkin type IGA is the minimization of the computational effort for numerical quadrature. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal design of double layer scallop domes using genetic algorithm

An efficient methodology is proposed for optimal design of large-scale domes with various topologies and dimensions in plan. The major concern with the optimal design of large domes is the difficulties arising from plurality of design variables, i.e., size and shape variables. This complexity has propounded the optimal design problem of large scale domes as a great challenge over the years. Thus, in current study, extending the novel idea of using parametric mathematical functions, design variables are correlated to the geometrical properties of domes through a new point of view. Additionally, a modified sizing approach is taken up while treating with element sections. In this way, the number of design variables is decreased. Consequently, fewer numbers of these variables provides an impressive condition that considerably takes down the computational efforts needed to explore the design space for finding the solution of optimization problem. Optimization task is performed by the robust technique of genetic algorithm. The presented approach is applicable to a wide variety of enormous domes with outsized number of nodes and members. However, to show applicability as well as computational advantages of the presented algorithm, a numerical example of scallop domes is investigated.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

Patch coupling with the isogeometric dual mortar approach

The use of a common set of basis functions for design and analysis is the main paradigm of isogeometric analysis. The characteristics of the commonly used non-uniform rational B-splines (NURBS) surfaces require methods to handle non-conforming meshes to attain an efficient computational framework. The isogeometric mortar method uses constrained approximation spaces to enforce a coupling of deformations at the interface between patches in a weak manner. This method neither requires additional degrees of freedom nor the choice of empirical parameters. The main drawback of the standard isogeometric mortar approach is the non-local support of the mortar basis functions along the interface. This yields a large number of nodes per element for elements adjacent to the interface. Thus, the computational costs increase significantly for mesh refinement. This issue is remedied by the use of dual basis functions for the mortar method, which is referred to as dual mortar method. In this contribution several choices for the dual basis functions for B-splines are proposed and compared. A special focus is set on the support of the dual basis functions and on the support of the resulting mortar basis functions. Numerical examples show the influence of the choice for the dual basis functions on the accuracy of the global stress distribution, on the fulfillment of the interface conditions and on numerical efficiency. The use of approximate dual basis functions is shown to be competitive to computations of conforming meshes in terms of accuracy and efficiency. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and Application of the Rational B-Spline Finite Element Method in 2D

Non-Uniform Rational B-Splines (NURBS) are basis functions used in CAD software to describe exact geometric models. The implementation of these basis functions in the context of the Finite Element Analysis (FEA) is known as isogeometric analysis. The concept and definition of NURBS is briefly presented here. Since these functions are implemented as shape functions for the isogeometric analysis, the refinement strategies are discussed. The example of an infinite plate with circular hole serves as a benchmark. Finally, isogeometric analysis is applied to gradient elasticity since NURBS functions are of higher continuity and this is required in gradient elasticity. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

绿色建筑前期设计阶段的多目标优化及多属性决策模型

综合考虑建筑物的体型参数、围护结构参数和功能布局的影响,运用层次分析法将描述性的功能目标转化为定量值,建立绿色建筑前期设计阶段的能耗、成本和功能的多目标优化模型。针对模型变量的离散性,以邻域拓扑结构改进粒子群算法,防止陷入局部最优,得到绿色建筑方案的Pareto解集。在绿色建筑多属性决策中引入马氏距离与组合赋权方法,对最优方案进行排序决策。通过案例分析验证该模型的效果,在保证一定功能的前提下,可获得较低的能耗和成本,实现绿色建筑设计理念。     

Adaptive extended isogeometric analysis based on PHT-splines for thin cracked plates and shells with Kirchhoff–Love theory

In this paper, a posteriori error estimation and mesh adaptation approach for thin plate and shell structures of through-the-thickness crack is presented. This method uses the extended isogeometric analysis (XIGA) based on PHT-splines (Polynomial splines over Hierarchical T-meshes), which is abbreviated as XIGA-PHT. In XIGA-PHT, the isogeometric displacement approximation is locally enriched with enrichment functions, which efficiently capture the displacement discontinuity across the crack face as well as the stress singularity in the vicinity of the crack tip. On the one hand, the rotational degrees of freedom (RDOFs) are not required in Kirchhoff–Love theory, which drastically reduces the complexity of enrichment mode and computational scale for crack analysis. On the other hand, the PHT-splines basis functions can automatically satisfy the requirement of C¹-continuity for the Kirchhoff–Love theory. Moreover, the PHT-splines facilitate the local refinement, which is the deficiency of NURBS-based isogeometric formulations. The local refinement is highly suitable for adaptive analysis. The stress recovery-based posteriori error estimator combined with the superconvergent patch recovery (SPR) technique is used to evaluate the approximate local discretization error. A new strategy for selecting enriched recovered functions in the enriched areas was proposed. Special functions extracted from the asymptotic stress solutions are applied to obtain the recovered stress field in the enriched area. The results of stress intensity factors or J-integral values obtained by the adaptive XIGA-PHT are compared with reference solutions. Several thin plate and shell illustrative examples demonstrate the effectiveness and accuracy of the proposed adaptive XIGA-PHT.     

A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies: (I) case where the rigid body motions are known a priori

We describe in this Note a method for the numerical simulation of incompressible viscous flow around moving rigid bodies; we suppose the rigid body motions a priori known. The computational technique takes advantage of a time discretization by operator splitting à la Marchuk-Yanenko and of a finite element space discretization on a fixed mesh, to combine a Lagrange multiplier/fictitious domain treatment of the rigid body motions with an L²-projection technique, to force the incompressibility condition. The results of numerical experiments concerning flow around moving disks at Reynolds number of the order of 100 are presented.     

Shape-Aware Matching of Implicit Surfaces Based on Thin Shell Energies

A shape sensitive, variational approach for the matching of surfaces considered as thin elastic shells is investigated. The elasticity functional to be minimized takes into account two different types of nonlinear energies: a membrane energy measuring the rate of tangential distortion when deforming the reference shell into the template shell, and a bending energy measuring the bending under the deformation in terms of the change of the shape operators from the undeformed into the deformed configuration. The variational method applies to surfaces described as level sets. It is mathematically well-posed, and an existence proof of an optimal matching deformation is given. The variational model is implemented using a finite element discretization combined with a narrow band approach on an efficient hierarchical grid structure. For the optimization, a regularized nonlinear conjugate gradient scheme and a cascadic multilevel strategy are used. The features of the proposed approach are studied for synthetic test cases and a collection of geometry processing examples.     

On optimization of the RBF shape parameter in a grid-free local scheme for convection dominated problems over non-uniform centers

Global optimization techniques exist in the literature for finding the optimal shape parameter of the infinitely smooth radial basis functions (RBF) if they are used to solve partial differential equations. However these global collocation methods, applied directly, suffer from severe ill-conditioning when the number of centers is large. To circumvent this, we have used a local optimization algorithm, in the optimization of the RBF shape parameter which is then used to develop a grid-free local (LRBF) scheme for solving convection–diffusion equations. The developed algorithm is based on the re-construction of the forcing term of the governing partial differential equation over the centers in a local support domain. The variable (optimal) shape parameter in this process is obtained by minimizing the local Cost function at each center (node) of the computational domain. It has been observed that for convection dominated problems, the local optimization scheme over uniform centers has produced oscillatory solutions, therefore, in this work the local optimization algorithm has been experimented over Chebyshev and non-uniform distribution of the centers. The numerical experiments presented in this work have shown that the LRBF scheme with the local optimization produced accurate and stable solutions over the non-uniform points even for convection dominant convection–diffusion equations.     

An improved isogeometrical analysis approach to functionally graded plane elasticity problems

The isogeometric analysis method is extended for addressing the plane elasticity problems with functionally graded materials. The proposed method which employs an improved form of the isogeometric analysis approach allows gradation of material properties through the patches and is given the name Generalized Iso-Geometrical Analysis (GIGA). The gradations of materials, which are considered as imaginary surfaces over the computational domain, are defined in a fully isoparametric formulation by using the same NURBS basis functions employed for the construction of the geometry and the approximation of the solution. The basic concept of the developed approach is concisely explained and its relation to the standard isogeometric analysis method is pointed out. It is shown that the difficulties encountered in the finite element analysis of the functionally graded materials are alleviated to a large degree by employing the mentioned method. Different numerical examples are presented and compared with available analytical solutions as well as the conventional and graded finite element methods to demonstrate the performance and accuracy of the proposed approach. The presented procedure can also be employed for solving other partial differential equations with non-constant coefficients.     

A new fictitious domain method in shape optimization

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from Mommer (IMA J. Numer. Anal. 26:503–524, 2006) to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method can be applied to three dimensional shape optimization problems.