Isogeometric topology optimization for continuum structures using density distribution function

This paper will propose a more effective and efficient topology optimization method based on isogeometric analysis, termed as isogeometric topology optimization (ITO), for continuum structures using an enhanced density distribution function (DDF). The construction of the DDF involves two steps. (1)  Smoothness: the Shepard function is firstly utilized to improve the overall smoothness of nodal densities. Each nodal density is assigned to a control point of the geometry. (2) Continuity: the high-order NURBS basis functions are linearly combined with the smoothed nodal densities to construct the DDF for the design domain. The nonnegativity, partition of unity, and restricted bounds [0, 1] of both the Shepard function and NURBS basis functions can guarantee the physical meaning of material densities in the design. A topology optimization formulation to minimize the structural mean compliance is developed based on the DDF and isogeometric analysis to solve structural responses. An integration of the geometry parameterization and numerical analysis can offer the unique benefits for the optimization. Several 2D and 3D numerical examples are performed to demonstrate the effectiveness and efficiency of the proposed ITO method, and the optimized 3D designs are prototyped using the Selective Laser Sintering technique.     

An isogeometric analysis approach to gradient damage models

Continuum damage formulations are commonly used for the simulation of diffuse fracture processes. Implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models. The C⁰‐continuity of traditional finite elements has hindered the study of higher order gradient damage approximations. In this contribution we use isogeometric finite elements, which allow for the construction of higher order continuous basis functions on complex domains. We study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations. Copyright © 2011 John Wiley & Sons, Ltd.     

An isogeometric approach to topology optimization of multi‐material and functionally graded structures

A new isogeometric density‐based approach for the topology optimization of multi‐material structures is presented. In this method, the density fields of multiple material phases are represented using the isogeometric non‐uniform rational B‐spline‐based parameterization leading to exact modeling of the geometry, removing numerical artifacts and full analytical computation of sensitivities in a cost‐effective manner. An extension of the perimeter control technique is introduced where restrictions are imposed on the perimeters of density fields of all phases. Consequently, not only can one control the complexity of the optimal design but also the minimal lengths scales of all material phases. This leads to optimal designs with significantly enhanced manufacturability and comparable performance. Unlike the common element‐wise or nodal‐based density representations, owing to higher order continuity of density fields in this method, their gradients required for perimeter control restrictions are calculated exactly without additional computational cost. The problem is formulated with constraints on either (1) volume fractions of different material phases or (2) the total mass of the structure. The proposed method is applied for the minimal compliance design of two‐dimensional structures consisting of multiple distinct materials as well as functionally graded ones. Copyright © 2016 John Wiley & Sons, Ltd.     

Weakening the tight coupling between geometry and simulation in isogeometric analysis: From sub‐ and super‐geometric analysis to Geometry‐Independent Field approximaTion (GIFT)

This paper presents an approach to generalize the concept of isogeometric analysis by allowing different spaces for the parameterization of the computational domain and for the approximation of the solution field. The method inherits the main advantage of isogeometric analysis, ie, preserves the original exact computer‐aided design geometry (for example, given by nonuniform rational B‐splines), but allows pairing it with an approximation space, which is more suitable/flexible for analysis, for example, T‐splines, LR‐splines, (truncated) hierarchical B‐splines, and PHT‐splines. This generalization offers the advantage of adaptive local refinement without the need to reparameterize the domain, and therefore without weakening the link with the computer‐aided design model. We demonstrate the use of the method with different choices of geometry and field spaces and show that, despite the failure of the standard patch test, the optimum convergence rate is achieved for nonnested spaces.     

T‐spline finite element method for the analysis of shell structures

A T‐spline surface is a nonuniform rational B‐spline (NURBS) surface with T‐junctions, and is defined by a control grid called T‐mesh. The T‐mesh is similar to a NURBS control mesh except that in a T‐mesh, a row or column of control points is allowed to terminate in the inner parametric space. This property of T‐splines makes local refinement possible. In the present study, shell formulation based on the T‐spline finite element method (FEM) is presented. Shell formulation based on NURBS or T‐splines has fundamental limitations because rotational DOFs, which are necessary in the shell formulation, cannot be defined on control points. In this study, the simple mapping scheme, in which every control point is mapped into one geometric point on the surface, is employed to eliminate the limitations. Using this mapping scheme, T‐spline FEM can be easily extended to the analysis of shells. The proposed shell formulation is verified through various benchmarking problems. This study is a part of the efforts by the authors for the integration of CAD–CAE processes. Copyright © 2009 John Wiley & Sons, Ltd.     

Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

A large deformation frictional contact formulation using NURBS‐based isogeometric analysis

This paper focuses on the application of NURBS‐based isogeometric analysis to Coulomb frictional contact problems between deformable bodies, in the context of large deformations. A mortar‐based approach is presented to treat the contact constraints, whereby the discretization of the continuum is performed with arbitrary order NURBS, as well as C⁰‐continuous Lagrange polynomial elements for comparison purposes. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. Results of lower quality are obtained from the Lagrange discretization, as well as from a different contact formulation based on the enforcement of the contact constraints at every integration point on the contact surface. Copyright © 2011 John Wiley & Sons, Ltd.     

Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners

The mapping method was introduced in Jeong et al. (2013) for highly accurate isogeometric analysis (IGA) of elliptic boundary value problems containing singularities. The mapping method is concerned with constructions of novel geometrical mappings by which push‐forwards of B‐splines from the parameter space into the physical space generate singular functions that resemble the singularities. In other words, the pullback of the singularity into the parameter space by the novel geometrical mapping (a non‐uniform rational basis spline (NURBS) surface mapping) becomes highly smooth. One of the merits of IGA is that it uses NURBS functions employed in designs for the finite element analysis. However, push‐forwards of rational NURBS may not be able to generate singular functions. Moreover, the mapping method is effective for neither the k‐refinement nor the h‐refinement. In this paper, highly accurate stress analysis of elastic domains with cracks and ∕ or corners are achieved by enriched IGA, in which push‐forwards of NURBS via the design mapping are combined with push‐forwards of B‐splines via the novel geometrical mapping (the mapping technique). In a similar spirit of X‐FEM (or GFEM), we propose three enrichment approaches: enriched IGA for corners, enriched IGA for cracks, and partition of unity IGA for cracks. Copyright © 2013 John Wiley & Sons, Ltd.     

X‐FEM in isogeometric analysis for linear fracture mechanics

The extended finite element method (X‐FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X‐FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X‐FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X‐FEM with conventional finite elements of equal degree, the NURBS‐based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree. Copyright © 2011 John Wiley & Sons, Ltd.     

Evolutionary topology optimization using design space adjustment based on fixed grid

Design space optimization for topology based on fixed grid is proposed and its superiority to conventional topology optimization is shown. In the conventional topology optimization, the design domain is fixed. It is, however, desirable to make the design domain evolve into a better one during optimization process by increasing or decreasing the number of design pixels or variables, which we call design space optimization. A breakthrough in obtaining sensitivities when design space expands has been made recently with necessary mathematical background, but due to coupling effect and others, sensitivity results have not been satisfactory. Three innovative implementations are developed in this paper. Firstly, the proper characteristics of artificial material are defined. The second one is to decouple neighbouring elements for exact design space sensitivities. The previous design space optimization has been tedious because only one layer can be added. So, the third innovation is a new expansion strategy with multi‐layers based on design space sensitivities. As a result, the proposed evolutionary method can get an optimum much faster than ever before especially for large‐scale problems. It is also conjectured that this gives higher probability of getting the global optimum, as confirmed by numerical examples. Copyright © 2005 John Wiley & Sons, Ltd.     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.     

An isogeometric extension of Trefftz method for elastostatics in two dimensions

In this paper, an approach to blend the Hybrid‐Trefftz Finite Element Method (HTFEM) and the Isogeometric Analysis (IGA) called the Isogeometric Trefftz (IGAT) method is presented. The structure of the isogeometric extension of the Trefftz method is formally the same as for its conventional counterpart, except the approximation of the boundary displacements and geometry that are carried out using the Non‐Uniform Rational B‐Splines (NURBS) instead of polynomials. In other words, only the element boundaries are approximated using NURBS basis while the Trefftz approximation is used in the interior of the elements. For that reason, IGAT can be ranked alongside recently developed Isogeometric Boundary Element Method (IGABEM), the NURBS‐Enhanced Finite Element Method (NEFEM), the Isogeometric Local Maximum Entropy (IGA‐LME) method, and the Isogeometrically enhanced Scaled‐Boundary element method (SBFEM), which all use NURBS approximation at the domain boundary only. Theoretical conjectures made in this paper are accompanied by three examples that show that IGAT leads to excellent results using only a few elements.     

Voigt–Reuss topology optimization for structures with nonlinear material behaviors

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance. © 1997 John Wiley & Sons, Ltd.     

Extended isogeometric analysis based on PHT‐splines for crack propagation near inclusions

Adaptive local refinement is one of the main issues for isogeometric analysis (IGA). In this paper, an adaptive extended IGA (XIGA) approach based on polynomial splines over hierarchical T‐meshes (PHT‐splines) for modeling crack propagation is presented. The PHT‐splines overcome certain limitations of nonuniform rational B‐splines–based formulations; in particular, they make local refinements feasible. To drive the adaptive mesh refinement, we present a recovery‐based error estimator for the proposed method. The method is based on the XIGA method, in which discontinuous enrichment functions are added to the IGA approximation and this method does not require remeshing as the cracks grow. In addition, crack propagation is modeled by successive linear extensions that are determined by the stress intensity factors under linear elastic fracture mechanics. The proposed method has been used to analyze numerical examples, and the stress intensity factors results were compared with reference results. The findings demonstrate the accuracy and efficiency of the proposed method.     

Mixed isogeometric analysis of strongly coupled diffusion in porous materials

In this paper, we develop a mixed isogeometric analysis approach based on subdivision stabilization to study strongly coupled diffusion in solids in both small and large deformation ranges. Coupling the fluid pressure and the solid deformation, the mixed formulation suffers from numerical instabilities in the incompressible and the nearly incompressible limit due to the violation of the inf‐sup condition. We investigate this issue using subdivision‐stabilized nonuniform rational B‐spline (NURBS) elements, as well as different families of mixed isogeometric analysis techniques, and assess their stability through a numerical inf‐sup test. Furthermore, the validity of the inf‐sup stability test in poromechanics is supported by a mathematical proof concerning the corresponding stability estimate. Finally, two numerical examples involving a rigid strip foundation on saturated soil and a swelling hydrogel structure are presented to validate the stability and to demonstrate the robustness of the proposed approach.     

Explicit level‐set‐based topology optimization using an exact Heaviside function and consistent sensitivity analysis

This paper presents a level‐set‐based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest‐descent update of the design variables in a level‐set method; the level‐set nodal values. An exact Heaviside formulation is used to relate the level‐set function to element densities. The level‐set function is not required to be a signed‐distance function, and reinitialization is not necessary. Using this approach, level‐set‐based topology optimization problems can be solved consistently and multiple constraints treated simultaneously. The proposed method leads to more insight in the nature of level‐set‐based topology optimization problems. The level‐set‐based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level‐set‐based or density‐based regularization can be used to avoid artifacts in the final results. The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level‐set‐based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Copyright © 2012 John Wiley & Sons, Ltd.     

Isogeometric topology optimization of anisotropic metamaterials for controlling high-frequency electromagnetic wave

This study presents a level set–based topology optimization with isogeometric analysis (IGA) for controlling high-frequency electromagnetic wave propagation in a domain with periodic microstructures (unit cells). The high-frequency homogenization method is applied to characterize the macroscopic high-frequency waves in periodic heterogeneous media whose wavelength is comparative to or smaller than the representative length of a unit cell. B-spline basis functions are employed for the IGA discretization procedure to improve the performance of electromagnetic wave analysis in a unit cell and topology optimization. Also, to keep the same order of continuity on the periodic boundaries as on other element edges in the domain, we propose the extended domain approach, while incorporating Floquet periodic boundary condition (FPBC). Two types of optimization problems are taken as examples to demonstrate the effectiveness of the proposed method in comparison with the standard finite element analysis (FEA). The optimization results provide optimized topologies of unit cells qualified as anisotropic metamaterials with hyperbolic and bidirectional dispersion properties at the macroscale.     

An isogeometric analysis Bézier interface element for mechanical and poromechanical fracture problems

Interface elements are a powerful tool for modelling discontinuities. Herein, we develop an interface element that is based on the isogeometric analysis concept. Through Bézier extraction, the novel interface element can be casted in the same format as conventional interface elements. Consequently, the isogeometric interface element can be implemented in a straightforward manner in an existing finite element software by a mere redefinition of the shape functions. The interface elements share the advantages of isogeometric continuum elements in that they can exactly model the geometry. On the other hand, they inherit the simplicity of conventional interface elements, but also some deficiencies, such as the occurrence of traction oscillations when a high interface stiffness is used. The extension towards poroelasticity is rather straightforward, and in this situation, the smoother flow profiles and the ensuing preservation of local mass balance are additional advantages. These are demonstrated at the hand of some example problems. Copyright © 2013 John Wiley & Sons, Ltd.     

A generalized finite element formulation for arbitrary basis functions: From isogeometric analysis to XFEM

Many of the formulations of current research interest, including iosogeometric methods and the extended finite element method, use nontraditional basis functions. Some, such as subdivision surfaces, may not have convenient analytical representations. The concept of an element, if appropriate at all, no longer coincides with the traditional definition. Developing a new software for each new class of basis functions is a large research burden, especially, if the problems involve large deformations, non‐linear materials, and contact. The objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis, resulting in a formulation that can be implemented within the traditional structure of a finite element program but that permits the use of arbitrary sets of basis functions that are defined only through the input file. Elements ranging from a traditional linear four‐node tetrahedron through a higher‐order element combining XFEM and isogeometric analysis may be specified entirely through an input file without any additional programming. Examples of this framework to applications with Lagrange elements, isogeometric elements, and XFEM basis functions for fracture are presented. Copyright © 2010 John Wiley & Sons, Ltd.